The German champions have reportedly agreed on a transfer worth about 40 million euros (NZ$67 million), including add-ons, for the 30-year-old, who had a year left on his Liverpool contract.
Mane leaves Anfield after 269 appearances, having scored 120 goals in all competitions, a stellar return that helped the club to the Champions League title in 2018-19 and the Premier League crown a season later.
He was among Liverpool's stand-out performers last season, netting 23 times as Juergen Klopp's side secured the domestic cup double and narrowly missed out on the league title to Manchester City.
Mane won six trophies in all following his switch from Southampton in June 2016, forming a potent attacking trio with Mohamed Salah and Roberto Firmino.
Klopp last month hailed Mane, his first major singing after taking over, as "world class" and likened him to a "machine" but the player, eager for a new challenge, declined a new contract.