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Kieran Gibbs current contract expires in the summer of 2018 and The Sun claim that Palace are lining up a move in the upcoming January transfer window.
Arsene Wenger is set to decide whether Gibbs will be kept for Arsenal’s title push, or if he will be sold before his contract runs down even further.
The left – back has also attracted interest from Swansea City.
The England International has featured just five times in the Premier League for Arsenal this campaign.
Palace are said to have been keeping tabs on the defender when Alan Pardew was in charge.
And it is claimed that they could test Arsenal’s resolve with a January bid, with Arsene Wenger aware his price will drop further if he does not sign a new deal.
Allardyce is not the only top flight boss alerted to Gibbs’ current situation at the Emirates.
However, with Bradley’s job far from safe, Gibbs could opt to move to Selhurst Park instead.
Swansea’s current plight means that the Arsenal defender could instead opt for the safety of Palace, who are expected to stay up with Allardyce in the job.