Mane became a prime target for Julian Nagelsmann's side, with Liverpool rejecting early offers for the forward, who had just 12 months to run on his contract at Anfield.
When Liverpool boss Jurgen Klopp brought in Benfica star Darwin Nunez for a reported fee of £64 million ($111.6 million), with a further £21.4 million ($37.3 million) in potential add-ons, it left the door open for Mane to depart.
Senegal coach Aliou Cisse encouraged Mane to join Bayern as "the best fit" for his career, and Nagelsmann's club has finally struck a deal to suit all parties.
Liverpool will reportedly receive an assured sum of €32 million ($47.7 million), plus €6 million ($8.9million) based on appearances and a further €3 million ($4.5 million) depending on future success that Mane and Bayern achieve.
Bayern's opening offer to Liverpool is understood to have been €25 million ($37.3 million) plus €5 million ($7.5 million) in add-ons that were all tied in to Mane and Bayern's success.
Mane scored 90 goals in 196 Premier League games for Liverpool after joining from Southampton in 2016.
Only Leicester City striker Jamie Vardy (104), Mane's Liverpool team-mate Mohamed Salah (118) and Tottenham Hotspur's Harry Kane (134) have managed more in the competition across that period.
Mane also played 51 games for Liverpool in the 2021-2022 season – only 10 players across Europe's top five leagues appeared more often – finding the net 23 times and assisting two more.
That was not enough to help Klopp's side to Premier League or UEFA Champions League success, but Mane did manage to lift the Carabao Cup and FA Cup trophies with the Reds in his final season at the club.
Mane won the Premier League and UEFA Champions League during his time with the Reds, as well as the FIFA Club World Cup and UEFA Super Cup.
His arrival in Bavaria leaves questions over the future of Bayern talisman Robert Lewandowski, who declared his story with Nagelsmann's side "over" as he pushes for a move to Barcelona.