The Ivory Coast international has been heavily linked with an exit from Selhurst Park this summer after scoring 10 Premier League goals last season.
Arsenal were initially considered the frontrunners to secure his signature but have since turned their attentions to Lille’s Nicolas Pepe.
Sky Sports News reports that the Toffees have come in with a cash-only £55m bid that doesn’t involve a player exchange element.
Zaha is understood to be willing to move to Goodison Park despite having preferred the option of remaining in London.
The report adds that Everton are also looking to sign Juventus striker Moise Kean and have entered talks for the Italy international.
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