The Senegal international, 24, is Jurgen Klopp’s top target following Mario Gotze’s decision to remain at Bayern Munich.
The Kop boss is expecting to land him but Starsport understands he could have to pay as much as £30m.
Klopp has followed Mane’s progress since his spell at Red Bull Salzburg and discussions with Southampton are ongoing.
Sane would be the latest in a long line of players to swap St Mary’s for Anfield.
Liverpool have signed Adam Lallana, Nathaniel Clyne, Dejan Lovren and Rickie Lambert from the South Coast club in recent seasons.