The Germany international was linked with a move to Bayern throughout the summer before staying put at the Etihad Stadium.
But the 23-year-old’s season was ruined by a cruciate ligament injury and he is yet to feature in the Premier League in 2019/20.
According to BILD, Sane made it clear that he wants to leave City next month and secure a return to the Bundesliga, where he starred for Schalke before making a £37 million move to Manchester in August 2016.
Sane isn’t expected to be back in action until at least February and it is unclear if Bayern will be prepared to make a move as early as January.
The German champions could also be put off by City’s £127 million demands for the forward, who has scored 39 goals in134 appearances for the club.
Sane scored 10 goals and contributed 11 assists in 31 Premier League appearances for Pep Guardiola’s side as they retained the Premier League title in impressive fashion last season.
But the form of Raheem Sterling and Bernardo Silva meant his game time was more limited than in previous years.
Sane scored 13 goals in 57 appearances during his spell at Schalke, and has five goals in 21 Germany caps.
His contract at the Etihad runs until the summer of 2021.
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