The Mirror claims that Zaha has rejected the offer of a contract extension with the Eagles, having signed a new deal just 15 months previously.
Everton were interested in signing the Ivory Coast international earlier this summer but Zaha was reluctant to move north again after his failed stint at Manchester United a few years ago, so the Toffees signed Richarlison from Watford instead.
It would require a huge offer to convince Palace to part with Zaha, without whom they are a significantly inferior team. However, he could lodge an official transfer request to help facilitate a move.
Palace's new contract offer would have seen Zaha's weekly wage rise from £100,000 to £125,000, but he wants to keep his options open.
Just last week, people jumped to conclusions after an Instagram post from Zaha hinted at a big life change. He clarified that it had nothing to do with football, but didn't rule out the possibility that he could be on the move before the transfer window closes on August 9.
It is thought that Chelsea want to sign Zaha as a pre-emptive replacement for Eden Hazard, who is keen on a move to Real Madrid. The Spanish transfer window closes three weeks after the English one so Chelsea don't want to lose Hazard without having already signed a replacement.
Zaha scored nine goals last season despite missing five weeks of the campaign with an injury. Palace lost all nine of the league matches for which their talismanic winger was absent.