The Bundesliga champions were keen on keeping the winger but Sane rejected the offer of a contract extension.
Bayern had included Sane in their 29-man squad for the Club World Cup, which begins on 15 June.
However, he is set to complete a deal with Galatasaray in the coming days, with the Athletic, external reporting he has verbally agreed a three-year deal with the Turkish club.
Sane, who won six major trophies in four years with City, claimed three consecutive Bundesliga titles during his first three seasons at Bayern before a fourth last term.
He scored 61 goals in 220 matches for Bayern, including 13 in 45 matches last season.
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