Earlier on Friday, reports emerged that Everton had entered talks with Palace with a view to making an offer of £60million plus a player for the forward.
Wilfried Zaha is also wanted by Arsenal, however is apparently willing to consider a move to Everton.
Everton are looking at offering James McCarthy and Cenk Tosun in a deal for Wilfried Zaha
Crystal Palace had an £8m offer for midfielder James McCarthy turned down last week
Everton emerged as a surprise front runner for the 26-year-old earlier in the week, indicating their opening offer would be £55million.
It would be a club record, topping the £45million they paid for Gylfi Sigurdsson in 2017.
Sources close to the negotiations say that they are prepared to go as high as £70million if they end up in a bidding war.
Everton are also offering striker Cenk Tosun in an attempt to get their man Wilfried Zaha
Crystal Palace have valued Wilfried Zaha at £80million, and have already rejected an offer of £45million from Arsenal this summer.
Palace had an £8m offer for McCarthy turned down last week, so including him in the deal for Zaha might be a way for Everton to get their man and keep both clubs happy.
Chelsea have also considered entering the race for the Ivorian, and are looking at the possibility of trying to sign him once they come through their transfer ban.