Marco Silva’s team joined Arsenal in the race for the Ivory Coast international at the weekend, although the Gunners' interest has subsequently cooled as they close in on signing Nicolas Pepe from Lille.
And while Everton have confirmed they did make an opening offer for the player, they have denied they returned with an improved £55m ($67m) follow-up bid that included two players in exchange.
"Everton categorically denies a bid has been submitted to Crystal Palace for Wilfried Zaha that included an increased sum plus Cenk Tosun and James McCarthy," the Toffees revealed in a statement to Sky Sports.
"Everton and Crystal Palace have very good relationship.
"A bid was made over weekend which was turned down. Both clubs agreed that was end of matter."
Former Everton boss Sam Allardyce, who also coached Zaha during his spell at Selhurst Park, says that, despite the club’s protestations, the transfer still seems like a good fit given Everton’s renewed ambition under owner Farhad Moshiri.
"I would have thought it was a possibility Everton would have tried to sign him because they’ve been one of the biggest players in the market since Farhad got control of Everton, and it looks like they’re splashing out again this summer," Allardyce told talkSPORT.
"Making an offer for Wilf is not beyond them, but whether they’ll reach [chairman Steve Parish’s] valuation, I’m not so sure.
"I thought the Arsenal move would have been the big one for Wilf, because he’d prefer to stay in London."
The Toffees have already completed the signings of Fabian Delph from Manchester City and goalkeeper Jonas Lossl from Huddersfield, as well as making the loan deal for Barcelona’s Andre Gomes a permanent signing.
Meanwhile French champions Paris Saint-Germain confirmed on Tuesday that they had signed Everton midfielder Idrissa Gueye in a reported £28m ($34m) deal ahead of the new season.