The Reds recently had a £10 million offer for the Southampton right-back rejected and are now reviewing their position whether to improve their offer.
Clyne signed for the Saints in 2012 from Crystal Palace and he has made 94 league appearances for the south coast club. He has just one year remaining on his contract at St. Mary's and despite the club hopeful he will ink a new deal, they are also prepared to sell Clyne if the right offer comes in.
The England international has also been a target for Manchester United, Manchester City and Chelsea so Liverpool could face stiff competition for his signature.
Clyne was a key member of the Southampton side who finished an over-achieving seventh in the Premier League this season, just one place behind Liverpool, and was also in many people's Team of the Year.
Brendan Rodgers looks to replace Glen Johnson's role in his side after the 30-year-old was not granted a new contract and will leave Liverpool on a free transfer.
Youngster Jon Flanagan is also set to miss the majority of next season after also missing the whole of last term with a knee injury, which he aggravated once again and Javier Manquillo has not been impressive since signing on a two-year loan deal from Atletico Madrid last summer.
Clyne showed his attacking capabilities throughout last season on the right-hand side of defence for Southampton and his qualities would suit Rodgers' attacking minded philosophy.
He scored a superb solo goal against the Reds on the opening day of this season at Anfield which would have impressed the Northern Irishman.
It's likely that Rodgers will not give up on Clyne's signature. Last summer, Liverpool chased Alberto Moreno for months and, after it looked like he would remain at Sevilla, the Spaniard eventually moved to the Merseyside club.