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Mane, 30, is expected to sign a three-year deal with the German champions, who are set to pay up to £35.1m for the forward.
Liverpool had previously rejected two bids from Bayern for Mane but have now agreed to sell the Senegal international, who had just one year remaining on his deal at Anfield.
According to The Athletic's David Ornstein, Mane will undergo a medical with Bayern next week before completing the move.
It brings to an end a hugely successful eight-year spell for Mane on Merseyside.
He first joined Liverpool in a £34m move from Southampton in 2016.
Mane would become a crucial player for the Reds, playing a major role in their success under Jurgen Klopp.
🚨 Sadio Mane will sign a three-year contract with Bayern Munich that runs until 2025 and he is scheduled to undergo a medical with the German champions next week before completing the transfer #FCBayern #LFC @TheAthleticUK https://t.co/wruATv4Z7W— David Ornstein (@David_Ornstein) June 17, 2022
Liverpool and Bayern Munich reach agreement on Sadio Mane deal. Guaranteed fee of £27.5million [€32million], plus £5million based on appearances and £2.6million based on individual and team achievements. The total package is worth £35.1million [€41million].— paul joyce (@_pauljoyce) June 17, 2022
He scored 120 goals in 269 games for the club, winning the Champions League, Premier League, Club World Cup, FA Cup and Carabao Cup during his time at Anfield.
However, he suggested he could be ready for a new challenge after Liverpool lost to Real Madrid in the Champions League final in Paris last month.
Speaking after the defeat, Mane said he would let fans in his home country decide his future amid speculation he was planning to leave Liverpool.
"I respect the fans and team but 60 to 70 per cent of Senegalese want me to leave Liverpool? I'll do what they want," he said.
He is now set to join Bayern and become their second major signing of the summer transfer window, following Ryan Gravenberch's €18.5m (£15.8m) move from Ajax.