The bid is being reported after news filtered out that their bid for Khouma Bacabar was unsuccessful.
Palace were hoping to make Bacabar a major January acquisition, but they are finding it difficult to reach an agreement with his club Fiorentina.
Barnes is a possibility in this window, either on loan or as a permanent move. Finances are not unlimited at Palace, but manager Roy Hodgson does have around £12 million that he can spend on players in the remaining days of the transfer window.
Palace will have to make a good bid in order to convince Burnley to let the player go, given he has 18 months left on his contract.
The fellow Premier League club would rather lose the player in the summer, when they have more time to get a replacement. A sale now would give them mere days to find a new striker.