Reds boss Jurgen Klopp wants to boost his attacking options and is willing to make Mane one of the biggest signings in the club's history to do it.
Liverpool and Southampton will continue to hold talks over the 24-year-old, with the fee a possible sticking point.
The Saints are likely to want around £40m for a player who scored 11 goals in 37 Premier League games last season.
This is around £10m more than the top end of Liverpool's price range, so there is still more negotiating to be done before any deal can be concluded.
The process might also be delayed by the fact that Southampton are still without a manager following the departure of Ronald Koeman to Everton.
Klopp was impressed by Mane, heavily linked with Manchester United during Louis van Gaal's time at Old Trafford, when he scored twice as Southampton came from 2-0 down to beat Liverpool 3-2 at St Mary's on 20 March.
Alongside England striker Daniel Sturridge and Belgium youngster Divock Origi, Mane would add pace and a goalscoring threat.
Liverpool would seek to recoup much of the money they spend on the Senegal international by selling striker Christian Benteke, who has struggled after his £32m switch from Aston Villa.
If he makes the move, Mane will be treading a well-worn path between Southampton and Liverpool.
Adam Lallana, Dejan Lovren, Rickie Lambert and Nathaniel Clyne have all moved to Anfield since the summer of 2014.
It would also give Saints a handsome profit on a player they bought from Salzburg for £10m two years ago.
It would also make Mane the third biggest signing in Liverpool's history behind Benteke and Andy Carroll, who came from Newcastle United in a £35m deal.
Southampton have also sold midfield man Victor Wanyama to Tottenham for £11m at the start of what could be another summer of transition at St Mary's.
Liverpool have made Southampton striker Sadio Mane a prime transfer target this summer and are willing to pay £30m.