I heard someone say that the Buddha is omnipresent. While this does not seem correct since the Buddha passed away, I recall hearing that the Buddha is an embodiment of the Dhamma (which could be considered omnipresent within conditioned existence). In which text was this stated?
In SN 22.87 the states, “Enough, Vakkali! Why do you want to see this foul body? One who sees the Dhamma sees me; one who sees me sees the Dhamma. For in seeing the Dhamma, Vakkali, one sees me; and in seeing me, one sees the Dhamma.”
Lots has been said about this line, but I think what is most important is that the individual realizes that a Buddha is perfected to the full extent, and his knowledge and vision exactly mirror the teaching he has described. So much so that there will be no discrepancies, and that if a person wants to understand what that means they have to “see” the Dhamma for themselves.
I think it is a mistake to interpret this as omnipresent, which is too easily mistaken for - as you say - some enduring existence of the Buddha. The endurance is only to the extent that the path to seeing the Dhamma is still available, and one can directly know that if they put forth the effort.