The German side are understood to have made a bid totalling £25million, an offer that was swiftly rejected by the Premier League club, however it's expected that they'll return with a new approach.
Speculation has been building around Mane's future with the Senegal international having just one year left on his Liverpool contract, and according to ex-Reds star Hamann, it seems that a move to the German champions is edging closer with Robert Lewandowski's move to Barcelona also gathering pace.
Speaking to talkSPORT Drive: "Yeah, what I hear is that he wants to come to Munich.
"He's made it pretty clear that he thinks his time in Liverpool is up and he wants to come and now it's a matter of how much Liverpool will want for him.
"Obviously Robert Lewandowski wants to join Barcelona, he only has a year left on his contract, so I think Bayern Munich will want to get both deals done in the next few days or next few weeks, but I think it's very likely that he plays his football here next year."
Hamann added: "If Lewandowski goes they obviously have to bring in a centre forward as a younger player.
"Saša Kalajdžić at Stuttgart is someone who has done well, I think they're after him as although Mane has played that role through the middle, I don't really like him there and I think he's a lot more effective on that left side.
"There's a question mark over whether Serge Gnabry's going to stay and Leroy Sane who has been disappointing over the last six months, so Bayern Munich could really do with him and I think he's a superstar.
"He's a world class player and the German league has developed world class players like Sancho, Dembele, Aubameyang, Lewandowski, but I think he's the first world class player in about 15 or 20 years to join the Bundesliga."
As far as Liverpool are concerned, Hamann believes that the Reds could face something of a dilemma given the identical situation surrounding Mohamed Salah's future.
He said: "It's tough because they've got a disciplined wage structure and if you look at how much they pay them, they could probably earn more money.
"To get a Mane or a Salah now, which they did five or six years ago, it's probably not possible anymore because there's too many clubs who will pay those wages and at the moment Liverpool doesn't.
"I don't see that changing with regards to Mane because all the talk in the last few months is about Salah's contract, and nobody talked about Mane even though he's only got a year left as well.
"If they were to lose both I don't think it would be very good for Liverpool.
"I think that Salah will stay, but if he can earn 50% more or twice as much somewhere else then obviously he will listen to other offers.
"If Mane leaves now, I think that Liverpool know they have to do everything they can to keep Salah."