Mane has been a key part of Liverpool's attack under manager Jurgen Klopp
The Reds have been left bemused after the German club returned with the offer after an initial bid, reported to be worth up to £25m, was turned down.
Liverpool are understood to want about €50m (£42.8m) for 30-year-old Mane considering Bayern's similar valuation of 33-year-old Robert Lewandowski.
Mane and Lewandowski both have one year remaining on their current deals.
Poland international Lewandowski wants to move to Barcelona this summer after a decade with Bayern.
Mane has been more coy about his future. The forward joked at the weekend that he would be led by public opinion in his homeland, suggesting the majority of Senegalese people would prefer him to leave Anfield.
But the forward has previously hinted that he will stay at Liverpool, promising a "special" answer over his future.
Fans pleaded with Mane to stay at the club during Liverpool celebrations for this season's FA Cup and Carabao Cup victories
Mane has scored 120 goals for Liverpool since arriving from Southampton for a fee of £36m in June 2016, helping the club to success in the Champions League in 2019 and the Premier League in 2020.
He signed a contract extension with the club in 2018, which expires at the end of next season.
Liverpool also face a conundrum over the future of Egyptian star Mohamed Salah, who will also be a free agent in summer 2023.
Salah has said he intends to remain at Liverpool next season but Barcelona are rumoured to want to sign him as a free agent at the end of the coming campaign.
On Wednesday, Liverpool were linked with a bid for 22-year-old Uruguay and Benfica striker Darwin Nunez.
-- Courtesy of BBC Sport