The Reds will receive 32m euros (£27.4m) with an additional 6m euros based on appearances, plus 3m euros for individual and team achievements.
Liverpool turned down two bids from Bayern before agreeing a fee for the Senegal forward, who was under contract until next summer.
“This is the right time for this challenge,” said the 30-year-old.”I felt the great interest of this big club from the start so there was no doubt in my mind.
“Mane joined Liverpool for £34m from Southampton in June 2016, and scored 120 goals in 269 games, finishing last season with 23 goals in all competitions.
“Sadio Mane is a world star who underscores the appeal of FC Bayern and increases the attractiveness of the entire Bundesliga,” said Bayern president Herbert Hainer.
“It is for such unique footballers that the fans come to the stadiums.”
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