Mane has now settled on a three-year contract with Bayern with just 12 months run on his Reds deal.
BILD says Bayern are aiming to make it third time lucky with a £34.6m bid. Despite still falling £7.4m short of the Reds' valuation, Bayern hope that their latest package will tempt Liverpool to sell.
The deal would see that Bavarian outfit pay £30.3m up front with £4.3m in add-ons. Mane's agent Bjorn Bezemer met with Bayern sporting director Hasan Salihamidzic in early May to discuss a move.
Reds boss Jurgen Klopp will not stand in the way of Mane's departure should both clubs agree a price.