The 26-year-old has a contract until the end of the current campaign with the Bhoys, and he has been in talks with club officials over a new deal.
However, Daily Express understands that the negotiations are not showing any progress, meaning that Ledley looks likely to leave in the near future.
Should he decide to leave the Bhoys, there will be plenty of Premier League sides ready to offer him an escape route from Parkhead, with Everton and Cardiff reportedly topping the list of suitors.
Joe Ledley left boyhood club Cardiff in the summer of 2010 to join Celtic on a free transfer, and has since scored 18 goals in 95 SPL appearances.
Having only played for the Bluebirds in the Championship, the Wales international is likely to be interested in a switch to English Premier League.