Liverpool have already had an initial bid rejected for the right-back.
Liverpool, however, are said to be negotiating a price in between. Southampton, however, know Liverpool's need is greater than Liverpool's given the right-back options at the disposal of both clubs.
Glen Johnson has left the club since his contract expired and Liverpool are short in that area. Javier Manquillo is possibly set to be offloaded after a difficult season, while Jon Flanagan's injury struggle means he won't be available until the start of 2016 if they're lucky.
Southampton seem to have prepared for the departure of Clyne, who has one year remaining on his contract at St Mary's, by securing the signing of Cedric Soares.
Liverpool will be keen to get things done, and it will seem odd if they try bargaining over a difference in fee they can easily make up in order to address what has been a problem position for a good few years.
The Reds have been pretty quick out of the blocks with their transfer business this summer, but ahead of a crucial season for the club where they really could do with hitting the ground running, they really shouldn't be allowing this to become a protracted affair, and should play ball with Southampton in order to get their man.