The German side now need to agree a fee with the Reds, after having a bid totalling £25million swiftly rejected earlier this week.
However, a new bid is expected, with Jurgen Klopp's side said to want around £40m for their forward.
Mane, 30, has one year remaining on his current Reds deal and comments from the attacker have hinted at an Anfield departure.
And Guardian journalist Fabrizio Romano reveals that a deal is close.
"Sadio Mane has reached full verbal agreement with Bayern on personal terms," he tweeted.
"Three-year contract ready - talks will enter into final stages with Liverpool, new bid set to be submitted.
"[Liverpool] are aware of his decision: Mane wants Bayern - and it's really close now."
Liverpool signed Mane's replacement on Tuesday, as Benfica striker Darwin Nunez was confirmed in a deal that could reach £85million.
That fee could make the Uruguayan forward the most expensive signing in Liverpool's history.