The Reds are believed to have identified Mane as their new primary target after their failed pursuit of Bayern Munich star Mario Gotze, with the Senegalese star having scored 25 goals in 75 appearances in his two seasons at the Saints.
With Ronald Koeman likely to be desperate to hold on to one of his side's most prized assets, it is thought that it would take a bid of at least around £25 million for the south coast club to consider selling the 24-year-old.
However, the Times claims that Jurgen Klopp would be prepared to offer Southampton a chance to bring reported target Jordon Ibe to St. Mary's as a key incentive in a potential deal.
Ibe has registered four goals and four assists this season but has been used almost exclusively as an impact substitute or a rotation starter, and Liverpool would likely be willing to part with the 20-year-old if it meant signing Mane became a more realistic proposition.
The Daily Star claims that Ibe has become frustrated with his inability to cement a regular place in Klopp's side and would be happy to continue his career elsewhere.
If a deal can be struck for Mane which includes a still-hefty price tag and Ibe's move, it could prove to be a profitable agreement for both sides.