Well I don't know the history of mediaeval Greek, but I'm guessing that the classical Greek name Konstantinopolis had changed to Stanbul in the Greek of 1453. The Turks then adopted this unchanged except for their insertion of the epenthetic i, and it has not changed in Turkish since.

The form 'Stamboul' formerly seen sometimes in English suggests the assimilation of nb to mb took place later in Greek, because if it was already Stambul in 1453 it would have been adopted as Istambul into Turkish. The Turkish cluster nb only makes sense if that's what it was when they borrowed it.

Now losing an initial syllable is easy, and so is dropping an ending. What remains is to explain how Stantinopol became Stanbul.

(Note names like Stavropol in Crimea, which to me looks obviously Greek, with loss of the ending. Was it a Greek colony conquered by the Russians at a time when -polis had become -pol?)

We stress the -NOP- but the loss of the -O- shows the stress was elsewhere in the Greek. Stantinopol became Stantinpol, perhaps losing two unstressed syllables to become Stantnpol. Obviously that awkward cluster -ntn- would reduce, giving Stanpol.

The voicing change giving Stanbol is a regular feature of modern Greek, and this shows it had already taken place by 1453.

Then I award myself one free sound change to get the u. Of course I'm making all this up as I go along, but it's a consistent story that fits all the facts I know. I expect something like this happened.