They have made an approach to Crystal Palace for the winger and indicated their opening offer would be £55million.
It would be a club record, topping the £45m they paid for Gylfi Sigurdsson in 2017.
Everton want to beat Arsenal to the signature of Crystal Palace star Wilfried Zaha this summer
Sources close to the negotiations have been assured Everton are serious challengers for Zaha and they are prepared to go as high as £70m if they end up in a bidding war.
Arsenal made the first move for wantaway Zaha with a £40m approach that was instantly dismissed for falling way below Palace’s valuation.
The Gunners considered increasing their transfer budget to fund an improved bid and also discussed offering players to Palace as makeweights, but they have failed to agree a deal with the south London club.
In an attempt to put off his suitors, Palace put a huge price tag on Zaha, demanding as much as £100m.
Palace have put a big price tag on the Ivory Coast international, demanding as much as £100m
While that fee would price Zaha out of a move, Palace chairman Steve Parish had told the 26-year-old the club would not stand in his way if a reasonable offer comes in.
Zaha wants to leave after a second spell at Selhurst Park — where he has spent a total of five years — and hopes to play Champions League football. Everton cannot offer that just yet, but owner Farhad Moshiri is willing to back manager Marco Silva to build a squad capable of challenging for the top four.
Meanwhile, midfielder Idrissa Gueye looks set to leave Goodison and join Paris Saint-Germain for £28m, although there are still details to be ironed out.
The 29-year-old, who has been one of the club’s best performers since his arrival in 2016, is open to a new challenge, particularly with the French club playing Champions League football.
Unai Emery is a huge admirer of Zaha but Arsenal have failed in their audacious pursuit so far