Sane’s current deal expires in 2021 and City have been trying to reach a new agreement for over a year, but a lack of progress has opened the door to Bayern and other suitors.
Bayern officials have confirmed their interest in bringing Sane to the Allianz Arena, although club president Uli Hoeness insists they will not pay more than €80 million (£71m/$90m) for any player. City are yet to receive any formal offers.
The Premier League champions would be willing to sell this summer, but only if the 23-year-old makes it known that he will not extend his contract, and if an offer of €100m (£88m/$112m) or more arrives.
City's preferred option is to keep the Germany international at the Etihad Stadium, however, and they will hold talks with Sane’s representatives - his father and a UK-based agent - in the coming days.
There is now some hope at the club that they will finally be able to reach an agreement and secure the winger’s future.
However, previous negotiations have not been straightforward, with Sane’s mother seen to be determined to derail talks and force a move to Bayern.
Sources have even told Goal that they believe stories linking Bayern with an interest in Sane have been leaked to the German media.
Should the upcoming discussions break down it would pave the way for Sane to leave the club, which would see City move for Benfica youngster Joao Felix.
City have laid the groundwork for a move for the 19-year-old forward, but would only push ahead with a formal offer if Sane were to move on.
Pep Guardiola has often been frustrated with Sane’s application in training and matches, but he has told his bosses that he wants to keep the former Schalke man as part of his squad for next season.
The Catalan says he tells his employers at the end of the season if there are any players who should be moved on, but he has stressed that Sane should be retained.