The England international has just a year remaining on his contract and has also reportedly been the subject of recent interest from Manchester United.
With Glen Johnson set to leave Anfield when his contract expires this summer and Jon Flanagan sidelined till the end of the year after further knee surgery, Liverpool have turned their attention to Clyne.
The 24-year-old joined Southampton from Crystal Palace in 2012 and made 41 appearances for Ronald Koeman???s team during the 2014/15 season.
Clyne earned his first England cap last November, playing the full 90 minutes in a 3-1 Euro 2016 qualifying victory over Slovenia, and has gone on to make four international appearances.
Liverpool currently have on-loan Atletico Madrid right-back Javier Manquillo in their ranks as well as Northern Ireland international Ryan McLaughlin.
Former England U21 international Andre Wisdom, who has spent the season on loan at West Brom, can also play at right-back.