The player is said to be frustrated after being promised Palace would accept reasonable offers this summer.
He wants to play Champions League football but understands he may not get that yet.
But Zaha feels the time is right for a new challenge.
The 26-year-old favours a move to Arsenal but Everton moved ahead of the Gunners in the race after the signing of Nicolas Pepe.
According to talkSPORT, Everton have now lodged a £30m bid for Alex Iwobi. Sky Sports have since confirmed the report and say the bid was rejected.
Everton have reportedly given up on Zaha and are now chasing an Arsenal alternative (Image: GETTY)
Everton have had a third bid for Zaha turned away by Crystal Palace (Image: GETTY)
Everton were looking to include Tosun in a deal for Wilfried Zaha (Image: GETTY)
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The Nigerian has played a big part under Unai Emery and it is not clear whether the Spaniard would be willing to sell.
Marco Silva wants to bring in another wide player to provide some more firepower in attack.
The signing of Moise Kean has sent a real statement of intent from Everton that they intend on challenging for Europe this season.
They are also working on a deal for Watford’s Abdoulaye Doucoure as they look to bolster their midfield ranks.