The 29-year-old Sane said farewell to Bayern fans in a video posted on social media on Thursday.
Sane, capped 70 times for Germany, has bagged 61 goals and registered 55 assists in 220 appearances for Bayern in all competitions.
He has won four Bundesliga titles, DFL-Supercups, a UEFA Super Cup and the Club World Cup with the German champions.
Sane joined Bayern in 2020 on a five-year deal from Manchester City, where he won the Premier League title in 2018 and 2019.