The Reds had previously lodged a £10million offer for the 24-year-old England international that was turned down straightaway by the Saints, who have valuated their speedstar at around £15 million.
Southampton are hoping to sign a new contract with Clyne, whose current contract runs out at the end of this next season.
Liverpool boss Brendan Rodgers wants to bring the Clyne to Anfiled as a replacement of Glen Johnson, who is out of contract this summer.
Manchester United are also interested in the England full-back but Liverpool look to be in pole position to secure his services.
The Saints are determined to persuade the defender to prolong his stay at St Mary. But if he refuses to re-sign, they will prefer to cash in and avoid losing him on free transfer next summer.
Clyne had joined Southampton from Crystal Palace in 2012 and made 94 appearances for the South Coast club.
He made his senior England debut in November last year and has earned five international caps to date.
With Clyne’s departure almost certain, Southampton have joined race to sign Burnley’s Kieran Trippier as a replacement.