Sane, whose City contract was due to expire at the end of next season, is set to sign a five-year deal with the German club.
City were resigned to selling the 24-year-old after manager Pep Guardiola confirmed the Germany international had turned down the Premier League club’s offers of a new contract “two or three times”.
Sane joined City from Schalke 04 in August 2016 for a fee reported by British media to be around 37 million pounds ($45.81 million) with performance-related add-ons.
The winger, who had 39 goals and 45 assists in 135 appearances in all competitions for City, was an integral part of their league title winning squads in 2017-2018 and 2018-2019.
However, he missed the majority of this season after suffering a cruciate ligament injury in the Community Shield match against Liverpool in August, making his comeback as a late substitute last week in a 5-0 win over Burnley.