Mane, 30, has become a firm favourite at Anfield over the last six years.
But he now embarks on a new challenge with the Bundesliga giants.
While the transfer has not yet been made official, Bayern director Hasan Salihamidzic is already looking forward to welcoming Mane aboard.
"Sadio is at the peak of his career, at a great age, is in great shape and he's greedy for titles - he's at exactly the right club," he told Bild.
"We're all proud and happy that such a world star comes to FC Bayern. Particularly because he was fired up for Bayern from the first minute. That means a lot to all of us - to me, the coaching team, the whole club.
"It's wonderful that we convinced such a player for our club."
In six seasons at Anfield the Senegal star netted 120 times in 269 games, while also lifting the Premier League, Champions League, FA Cup and Carabao Cup under Jurgen Klopp.
He has also enjoyed success at international level, winning the 2021 Africa Cup of Nations with Senegal.