Mane has just one year left on his contract at Jurgen Klopp's club and has been linked with an Anfield exit.
The 30-year-old is believed to be open to a move away and Bayern and Paris Saint-Germain are among the top clubs said to be interested.
The Times report that the Bundesliga champions made an offer worth a total of £25million, which was turned down.
It included a guaranteed initial fee of £21m, with £4m of add-ons, of which a large amount would depend on Bayern winning the Champions League.
Liverpool did not entertain the offer and are believed to have not given an indication of a price they would be prepared to sell at. Bayern are expected to make an improved offer.
Despite their unwillingness to entertain offers at this stage, the Reds have been linked with replacements.
Jurgen Klopp will want a fee for Mane that would allow him to find a replacement, with Benfica's Darwin Nunez in the running, according to The Times.
Nunez has been linked with several clubs, including Manchester United, after ending the season on 34 goals in 41 games, with Liverpool are also said to have a genuine interest in the 22-year-old.
The Reds know full well what he can do, too, with the the 22-year-old Uruguayan scoring in both legs of Benfica's Champions League quarter-final defeat to the eventual finalists.
Leeds star Raphinha and Chelsea's Christian Pulisic are among the Merseysiders' other reported targets.
Reds forward Mane has also been speaking about his future and said '60 to 70 per cent of Senegalese' people want him to leave the club after their Champions League final defeat, which he then hinted that he would do.
But, after scoring a hat-trick in Senegal's 3-1 win over Benin, he clarified his comments, insisting that he was 'joking' and he has a lot of 'respect' for Liverpool.
Speaking to Canal+, Mane said: "I spoke yesterday while joking with a bit of humour and it was everywhere. I think we'll stop there.
"Liverpool is a club I respect a lot. The fans adopted me since day one. Regarding the future, we'll see."