Ledley (R) celebrates Wales' opening day win against Slovakia
The Clarets are yet to add to their side since winning last season's Championship, despite being weakened by the loss of Joey Barton to Glasgow Rangers.
Barton has signed for Rangers on a free transfer
And, according to the Sun, Sean Dyche's men could move for the Wales star, as well as Stoke City midfielder Charlie Adam, as the 33-year-old's replacement.
Palace are said to value Ledley, for whom they paid £2.65 million (Transfermarkt) to sign from Celtic two-and-a-half-years ago, at around £4 million, despite the 29-year-old having only a year to run on his contract.
Earlier this month, the Sun reported that Palace were keen to offload fellow midfielder Jordon Mutch in a bid to raise funds to sign Newcastle United winger Andros Townsend.
Townsend is a reported Palace target
However, with Ledley having made just 11 starts during the Eagles' most recent Premier League campaign, it is feasible they could opt to sell the Cardiff-born midfielder instead.
Any decision on Ledley's future will likely be put on hold due to Wales' continued participation in the 2016 European Championship.