Last week, Everton submitted a massive £52m plus add-ons bid for the Ivory Coast international, but it was quickly rejected by the Eagles.
Having sold Aaron Wan-Bissaka to Manchester United for £50m earlier this summer, Palace are now reluctant to sell Zaha.
Initially, Palace were demanding a fee in the north of £80 million for the winger but now they have made it absolutely clear that it is pointless to come up with another offer.
Zaha is arguably one of the best wingers in the Premier League and enjoyed a fantastic 2018/19 campaign, where he scored 10 goals and provided five assists.
Palace now have no intentions of selling him this summer as they feel it is too late in the transfer window to replace him.
Everton boss Marco Silva is looking to bolster his attacking midfield region following the departure of Ademola Lookman, and has earmarked Zaha as his first choice target.
It remains to be seen whether Everton still make another bid for him in the coming days or Silva turns his attention to other available options.