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I am listening to a Dhammatalk where the Ajahn says ‘you don’t want to fall asleep when you die, because then you have no choice, you just go the way of your kamma. But if you have sati at the moment you die, then you have a choice where to go’. Do you please have some references on Suttas that talk about the dying process and your subsequent rebirth, and if possible which address this particular issue?
PS
The video in question is this, at about 8:55
AN4.171 might be helpful to study:
Mendicants, there are four kinds of reincarnation. –AN4.171
…
Sāriputta, take a person who has given up the lower fetters. In the present life they enter and abide in the dimension of neither perception nor non-perception.
They enjoy it and like it and find it satisfying.
If they abide in that, are committed to it, and meditate on it often without losing it, when they die they’re reborn in the company of the gods of the dimension of neither perception nor non-perception.
When they pass away from there, they’re a non-returner, not coming back to this state of existence.
Here it is in English (in current Sutta Central instead of Pali in the legacy version):
What an excellent choice to suggest!
The dying moment is something important, but to have sati at the moment of death isn’t an easy thing. It’s better to have been doing sati all of one’s life. In another sutta Mahanama who seems to be concerned with dying, is advised that how a person lives their life is much more important than what happens in the dying moment. Whether intentions come true, during the dying moment also depends on kamma.
That Mahanama sutta (SN55.22 I think) is a nice sutta (SN55.21 is very similar too). He is worried about dying suddenly without mindfulness. The simile of the inclining tree that the Buddha uses is quite apt. The sutta is not too long so I may as well cut as paste it here! 
So i have heard. At one time the Buddha was staying in the land of the Sakyans, near Kapilavatthu in the Banyan Tree Monastery. Then Mahānāma the Sakyan went up to the Buddha, bowed, sat down to one side, and said to him:
“Sir, this Kapilavatthu is successful and prosperous and full of people, with cramped cul-de-sacs. In the late afternoon, after paying homage to the Buddha or an esteemed mendicant, I enter Kapilavatthu. I encounter a stray elephant, horse, chariot, cart, or person. At that time I lose mindfulness regarding the Buddha, the teaching, and the Saṅgha. I think: ‘If I were to die at this time, where would I be reborn in my next life?’”
“Do not fear, Mahānāma, do not fear! Your death will not be a bad one; your passing will not be a bad one. A noble disciple who has four things slants, slopes, and inclines towards extinguishment. What four? It’s when a noble disciple has experiential confidence in the Buddha … the teaching … the Saṅgha … And they have the ethical conduct loved by the noble ones … leading to immersion.
Suppose there was a tree that slants, slopes, and inclines to the east. If it was cut off at the root where would it fall?”
“Sir, it would fall in the direction that it slants, slopes, and inclines.”
“In the same way, a noble disciple who has four things slants, slopes, and inclines towards extinguishment.”
This sutta called ‘sick’ is also good, in which the path to entering the stream, is discussed:
Thank you for your replies. These suttas however seem to indicate that what counts is how you live, and also that ideally one should aim for no rebirth at all rather than a good rebirth. I can’t find any teaching by the Buddha in these suttas according to which by being mindful at teh end of your life you can choose your rebirth.
The Devas say so to Citta in one of the suttas, but this does not mean that it is true, since in many cases in the suttas the Buddha shows that they are wrong.
Dear Friends in the Dhamma,
Here are my views on the question of Rebirth by Intention:
Lord Buddha has given guidelines on how to choose one’s rebirth. As mentioned by previous contributors, those guidelines must be learned, be trained gradually & practised, regularly, on a continuous basis.
One specific Sutta that gives such clear guidelines on choosing one’s rebirth is the Saṅkhārupapatti Sutta (Rebirth by Choice), MN 120. This is the fourth of the four Suttas in the Majjhima Nikaya that could be considered to be very relevant to our practice, (i.e. MN 117, 118, 119 & 120). The erudite members of the Sangha, (mostly of Ariya Sangha, in my view), who codified the Lord Buddha’s teachings into the main Nikayas, may have placed them in next to each other because of the connectivity & the importance that they have attached to these four Suttas.
Just for the record, the process of codification took place during various Sangha Councils, starting from the one that was chaired by the Great Arahant Maha Kassapa, soon after the Lord Buddha’s Final Passing Away & culminating in the writing of the Suttas into the Pali Cannon in Sri Lanka, about 1600 years ago.
Every Sutta in the Pali cannon is an important Sutta but, these four Suttas seem to have direct impact & connected guidance to help one’s practice.
Here is a section from (the first part of), the Saṅkhārupapatti Sutta:
Quote: “The Buddha said this:
“I shall teach you rebirth by choice. Listen and pay close attention, I will speak.
“Yes, sir,” they replied. The Buddha said this
“Take a mendicant who has faith, ethics, learning, generosity, and wisdom. They think: ‘If only, when my body breaks up, after death, I would be reborn in the company of well-to-do aristocrats!’
“They settle on that thought, stabilize it and develop it. Those choices and meditations of theirs, developed and cultivated like this, lead to rebirth there. This is the path and the practice that leads to rebirth there.” Unquote.
In the Saṅkhārupapatti Sutta, Lord Buddha discourses to reveal the various realms that one can aspire to be reborn. Here is the link to the Sutta from Suttacentral.
It is clear though, starting to practise these five factors i.e. faith, ethics, learning, generosity, and wisdom , at the 11th hour , just on one one’s death bed would not be enough at all but, would need to be practised over the entire lifetime.
These five factors cover the entire Noble Eightfold Path! So, there are no short cuts!
As a minimum prerequisite, one must be established in one’s Sila, the Ethics! Without the basis of ethics, the rest of the factors will not fall into their places. In the practice of the Dhamma, as a minimum,the ethical practice should be the one’s way of life. If one has been an ethical person for a long time then, one needs to continue in the same way. If not, one could forget the past, start from now & continue to be ethical for the rest of one’s life, until death. That should produce the desired results. To be clearer on this aspect of Kamma & its results, you may please refer to the Sutta titled Loṇakapalla Sutta (A Lump of Salt) AN 3.100. Here is the link:
One additional factor that the Lord Buddha has brought into the equation, within this Sutta, is the additional requirement of to be “settled on that thought, stabilize it and develop it”. So, one must be clear where one wants to be reborn, (Human Realm, or any of the many Deva Realms), so that the thought backed by the five factors (faith, ethics, learning, generosity, and wisdom) would take one to the desired destination.
This additional factor of clear intention appears to be a factor of prime importance. It does seem that this intention must be in the back of one’s mind when practising the Dhamma but, would need to be careful so that it doesn’t act as a hindrance to the practice that would lead to one’s final deliverance! Intention should be to end all suffering & finish the cycle of rebirth but, if that cannot be achieved then, one should have a clear intention as to where one would want to be reborn. That would be a half-way house to complete one’s practice to its final completion, the ending of all suffering!
So, if one could practise the five factors as detailed in the Saṅkhārupapatti Sutta & have the unwavering intention of wanting to be reborn say, for example, in the realm of the ‘The Gods of the Thirty-Three, (Tāvatiṃsa Deva Loka), one may reappear there after one’s death. All that you need to do is practise the Noble Eightfold Path!
One could start by being generous! If you want to help, there are so many ways to be generous. What is needed is the desire that the Lord Buddha called, ‘Chanda’, in Pali. If there is a will, there is a way, always!
May all being be well & happy!
With Metta,
Upasako
The below sutta extract is the closest thing I can think of in the suttas to wishing oneself to a specific rebirth:
“It’s by reason of this Dhamma conduct & harmonious conduct that some beings here, with the break-up of the body, after death, re-appear in a good destination, a heavenly world.
“Householders, if one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I, with the break-up of the body, after death, were to reappear among well-to-do noble warriors,’ it is possible that one—with the break-up of the body, after death—would appear among well-to-do noble warriors. Why is that? Because one is a person of Dhamma conduct, harmonious conduct.6
“If one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I, with the break-up of the body, after death, were to reappear among well-to-do brahmans,’ it is possible that one—with the break-up of the body, after death—would appear among well-to-do brahmans. Why is that? Because one is a person of Dhamma conduct, harmonious conduct.
“If one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I, with the break-up of the body, after death, were to reappear among well-to-do householders,’ it is possible that one—with the break-up of the body, after death—would appear among well-to-do householders. Why is that? Because one is a person of Dhamma conduct, harmonious conduct.
“If one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I, with the break-up of the body, after death, were to reappear among the Devas of the Four Great Kings,’ it is possible that one—with the break-up of the body, after death—would appear among the Devas of the Four Great Kings. Why is that? Because one is a person of Dhamma conduct, harmonious conduct.
“If one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I, with the break-up of the body, after death, were to reappear among the Devas of the Thirty-three… the Devas of the Hours… the Contented devas… the devas Delighting in Creation… the devas Wielding Power over the Creations of Others,’ it is possible that one—with the break-up of the body, after death—would appear among the devas Wielding Power over the Creations of Others. Why is that? Because one is a person of Dhamma conduct, harmonious conduct.7
“If one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I, with the break-up of the body, after death, were to reappear among the devas in Brahmā’s retinue… the Radiant devas… the Devas of Limited Radiance… the Devas of Immeasurable Radiance… the Devas of Streaming Radiance… the Beautiful devas … the Devas of Limited Beauty… the Devas of Immeasurable Beauty… the Beautiful Black devas… the Sky-fruit devas,’ it is possible that one—with the break-up of the body, after death—would appear among the Sky-fruit devas. Why is that? Because one is a person of Dhamma conduct, harmonious conduct.8
“If one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I, with the break-up of the body, after death, were to reappear among the Not Falling Away devas… the Untroubled devas… the Good-looking devas… the Clear-seeing devas… the Peerless devas,’ it is possible that one—with the break-up of the body, after death—would appear among the Peerless devas. Why is that? Because one is a person of Dhamma conduct, harmonious conduct.9
“If one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I, with the break-up of the body, after death, were to reappear among the devas who have reached the dimension of the infinitude of space… the devas who have reached the dimension of the infinitude of consciousness… the devas who have reached the dimension of nothingness… the devas who have reached the dimension of neither perception nor non-perception,’ it is possible that one—with the break-up of the body, after death—would appear among the devas who have reached the dimension of neither perception nor non-perception. Why is that? Because one is a person of Dhamma conduct, harmonious conduct.10
“If one—a person of Dhamma conduct, harmonious conduct—should wish: ‘O if I—with the ending of the effluents—were to enter & remain in the effluent-free awareness-release & discernment-release, having directly known and realized them for myself right in the here & now,’ it is possible that one—with the ending of the effluents—would enter & remain in the effluent-free awareness-release & discernment-release, having directly known and realized them for oneself right in the here & now.11Why is that? Because one is a person of Dhamma conduct, harmonious conduct.”
When this was said, the brahman householders of Sāla said to the Blessed One: “Magnificent, Master Gotama! Magnificent! Just as if he were to place upright what was overturned, to reveal what was hidden, to show the way to one who was lost, or to carry a lamp into the dark so that those with eyes could see forms, in the same way has Master Gotama—through many lines of reasoning—made the Dhamma clear. We go to Master Gotama for refuge, to the Dhamma, & to the Saṅgha of monks. May Master Gotama remember us as lay followers who have gone for refuge from this day forward, for life.”
Thank you @Polarbear and @Upasako for these good references. In both of them, as far as I understand, the two components are the good qualities of the person ( faith, ethics, learning, generosity, and wisdom in one sutta and Dhamma conduct, harmonious conduct in the other) and the wish (apparently a wish often formulated in one’s life) to be reborn in a given realm. Btw they don’t mention whether e.g. you can choose your family or your specific conditions if you are reborn as a human, just more general things like the caste.
But there does not seem to be any mentioning og sati at the time of death, nor of the fact that the wish should be formulated at that specific time, as far as I see.
Generally speaking, if it cannot be found in the Suttas, then it is reasonable to assume that it was added later on, within different practice traditions and/or commentaries.
There is much material available at S.C about this, given that the goal of SC is to identify, through scholarship, exactly what were ‘the words of the Buddha’ (Early Buddhist Texts) and texts that have come later. You can do a search through Essays and AUdio-visual resources, if you are interested in details 
ok thank you, so you mean to use the search at the top right of this page with some keywords, and then select AV and Essays to search from?
Yep 
A world of wonderful resources awaits
Enjoy!
My take in the suttas above is that just wishing for a good rebirth is not enough, it relies on harmonious conduct over a period of time. Bright kamma.
These suttas might help:
“Mendicants, when mindfulness of death is developed and cultivated it’s very fruitful and beneficial. It culminates in the deathless and ends with the deathless. And how is mindfulness of death developed and cultivated to be very fruitful and beneficial, to culminate in the deathless and end with the deathless?..AN6.20 SuttaCentral
“Mendicants, a mendicant should await their time mindful and aware. This is my instruction to you…
Suppose an oil lamp depended on oil and a wick to burn. As the oil and the wick are used up, it would be extinguished due to lack of fuel.In the same way, feeling the end of the body approaching, a mendicant understands: ‘I feel the end of the body approaching.’ Feeling the end of life approaching, they understand: ‘I feel the end of life approaching.’ They understand: ‘When my body breaks up and my life is over, everything that’s felt, since I no longer take pleasure in it, will become cool right here.’ SN36.8 SuttaCentral
A relevant Sutta;
“Householders, if both husband and wife wish to see one another not only in this present life but also in future lives, they should have the same faith, the same virtuous behavior, the same generosity, and the same wisdom. Then they will see one another not only in this present life but also in future lives.” https://suttacentral.net/an4.55/en/bodhi
I believe there a story about queen Mallika who had a bad thought at the end of her life and went to a bad birth, for 7 days. Because of the good kamma she had accumulated the rest of her life she became a deva afterwards! It seems the last moment is important but not in an overriding way. The Theravada tradition has made much of it, based on the Abhidhamma, however.