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The winger has outlined his plans to move this summer and hopes to secure a transfer before the squad returns for pre-season training next month.
Zaha has been priced out of a move in each of the past two summers but, while chairman Steve Parish does not want to sell, Palace could grant him his wish if a reasonable offer comes in because he has just two years left on his contract.
It is not clear yet what options would be open to Zaha. Everton have long been interested in the 28-year-old but Carlo Ancelotti’s departure to Real Madrid could change their transfer plans.
While a move may not happen, Palace want to be prepared should he leave and early discussions have been held with representatives of a number of forwards.
The Eagles have long-standing interest in former Chelsea winger Jeremie Boga, though it is unclear whether they could return for the Sassuolo man this summer.
A sticking point, as always, will be the fee Palace demand for Zaha.
The Ivory Coast international will always be more valuable to Palace than any other team would probably be willing to pay, and 25 per cent of any sale will go to Manchester United as part of the deal that brought Zaha back to Selhurst Park in 2015.
Palace, who are still searching for a new manager, will also be reluctant to lose Zaha because Eberechi Eze will miss the start of next season with an Achilles injury.
Palace are still working to appoint a new manager of their own. Talks have been held with former Wolves manager Nuno Espirito Santo, though he is also in the running for the Everton job.
Asked about the appointment of a new manager, Parish told the Daily Mail: "You need somebody who'll be a part of making something better by the time they've gone.
"There are trade-offs. We also realise we've got to reinvigorate the squad."
That desire to shake-up the squad this summer could prove to be a boost for Zaha. Should Roy Hodgson's successor be comfortable with their new star man leaving for an acceptable fee, the funds could be available to be pumped back into a new-look squad.
A cheaper option would be to retain a number of the players who are out of contract this summer, though that would not address the high average age of the group.
Christian Benteke has already sealed a new two-year deal, though most of those players out of contract at the end of this month are waiting for the new manager to make the final call on their futures.