The German international has signed a five-year contract with the Bundesliga champions after four years at the Etihad in a deal that could be worth up to £55million.
Bayern will pay an initial €49m (£44.7m) for Sané with another €11m (£10.1m) in add-ons, while City have also secured a 10 per cent sell-on clause.
Sané flew to Germany earlier this week to finish up the move but Uefa regulations mean he will not be allowed to play for his new club until next season.
Manager Pep Guardiola revealed earlier this month that Sané was set to leave City having rejected a new contract offer with no hopes of negotiations re-opening.
Sané’s existing contract was set to expire in the summer of 2021 and City were willing to let the winger enter the final year of his deal if Bayern failed to meet their asking price.
City are understood to believe that the potential £54.8m represents value for money for a player of Sané’s ability in the current transfer market, which has been significantly affected by the Covid-19 pandemic.
If Sané’s add-ons are achieved at Bayern, City will earn a profit of approximately £17.8m on a player who joined from Schalke 04 for £37m in 2016.
Sané had appeared to be nearing a move to Bayern last summer, only to suffer damaged cruciate ligaments in the Community Shield against Liverpool days before the close of the summer transfer window.
It is understood that Sané will earn approximately £386,000-a-week at Bayern – a salary that City were unwilling to match given the number of other options available to Guardiola in the same position.
Guardiola has suggested that replacing Sané will not be a priority for City during the next transfer window. “We have enough good players up front,” he said earlier this month.
“Gabriel [Jesus] can play on the left, Phil Foden can play on the left, we have Raheem [Sterling] there. We have big, talented players up front.
“Most of them are in good healthy conditions – like Phil, Riyad [Mahrez], Gabriel, Raheem – they can play quite regularly for a long period of time, they are so stable in terms of injury situations.”