The England international has fallen down the pecking order at Anfield after struggling with a back injury last season.
Clyne was loaned to Bournemouth in January and he impressed under Cherries boss Eddie Howe.
The 28-year-old is looking to move on this summer in order to preserve his chances of playing for the Three Lions and Roy Hodgson is ready to make his move.
Hodgson is bracing himself for bids for Aaron Wan-Bissaka and Clyne could well be his replacement at Selhurst Park.
West Ham United are also interested, while Napoli are pondering a move for the Liverpool ace, but whoever wins the race will need to pay at least £15million,
according to reports.
Clyne came through the ranks as a youngster at Palace and played 122 times for them in the Premier League between 2008-12 before joining Southampton.