The 24-year-old forward underwent a series of tests and scans at the Spire Liverpool Hospital in the city before sitting down with club officials to thrash out personal terms.
As a result, Mane's switch from Southampton will not be completed until Tuesday at the earliest although there is a chance discussions could drag things out until Wednesday.
The transfer fee is believed to be at least £30million which would make Mane one of the top three most expensive signings in Liverpool's history.
Currently, the Reds' record transfer is the £35million paid to Newcastle for Andy Carroll in 2011 while Christian Benteke was a £32.5m signing from Aston Villa last summer.
Senegal international Mane, who scored 25 goals in 75 appearances for the Saints, will become Jurgen Klopp's fourth signing after centre-back Joel Matip, midfielder Marko Grujic and goalkeeper Loris Karius.
Liverpool's players who have not been involved in the European Championship are due to return for pre-season training on Saturday, with their first friendly the following Friday.