Reports on Thursday claimed that the Ivory Coast international has asked to leave Selhurst Park admit interest from the Gunners.
Arsenal have been viewed as the leading candidates to sign Zaha this summer, despite having an opening £40 million bid rejected.
However, the Evening Standard believes that they could face competition from Premier League rivals Everton.
The Goodison Park club are set to bring in around £22.5m by selling Ademola Lookman to RB Leipzig and want to use that money to fund the signing of another winger.
Palace value the 26-year-old Zaha at £80m, but Toffees majority shareholder Farhad Moshiri has previously demonstrated that he is willing to splash out big sums in the market.
Everton could struggle to convince the player to join them, though, as he has spoken of his desire to play in the Champions League before and his brother Judicael Zaha has said that Arsenal is his preferred next destination.
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