Ledley, who has scored 18 goals in 95 appearances for Celtic, is out of contract in the summer and has been a key player under manager Neil Lennon.
Celtic chief executive Peter Lawwell confirmed in September that the club had opened talks with the former Cardiff ace, but the Daily Express are reporting that little headway has been made in convincing Ledley to sign a new deal with the club.
Ledley, who has made 46 appearances for Wales, could enter into pre-contract negotiations in January, and after leaving The Bluebirds for nothing in 2010, Celtic fear he could repeat that by leaving the Hoops at the end of his contract.
English Premier League sides Everton, and former employers Cardiff are both interested in the midfielder following a string of impressive performances for Celtic and Wales, and while Lawwell was keen to strike a quick deal with Ledley, the chances of him extending his stay appear slim.
Ledley hinted during the summer that he could make a decision on his future based on which players left the club, and with Victor Wanyama, Gary Hooper and Kelvin Wilson already departed, and Fraser Forster likely to be the next to go, the Welshman - who is one of the club’s highest earners - could exit Celtic in the summer months. The club’s wage constraints also mean the chance of a vastly improved deal is unlikely.
Celtic remain in talks with Georgios Samaras, whose contract is also up in the summer, while Portuguese cracks Benfica were linked with Forster in August.