Liverpool are in the market for a right-back after they decided to release Glen Johnson, whereas Manchester United are likely to be in search of someone more suited to that position than natural right winger Antonio Valencia.
Jon Flanagan is out for another nine months at Anfield due to a knee injury while Javier Manquillio, who is halfway through a two-year loan spell from Atletico Madrid, has found it difficult to settle in the Premier League.
Given that Clyne has one-year remaining on his contract with the Saints, Liverpool are hopeful of luring the 24-year-old to Merseyside, but a £10million bid has reportedly been rejected, as per the Telegraph.
If Manchester United are no longer interested in Clyne, Liverpool will become the front-runners for the right-back's signature - if they place a follow-up bid.
Southampton Director of football Les Reed previously sounded optimistic about the club's chances of persuading Clyne to extend his contact, as per talkSPORT.
Clyne joined Southampton three years ago from Crystal Palace, where he progressed through the ranks as a youth graduate. He has now made four senior appearances for England, while being selected for the upcoming friendly against the Republic of Ireland and a European Championship qualifier in Slovenia next month.