If he accepts any kind of contract offer, he's accepting he's an outfield version of a 3rd choice keeper - there to make up the numbers, maybe get the chance to start one full game a season IF LUCKY and a massive injury crisis occurs.
I'd be very surprised if another club would even come close to matching what he's probably been offered at Everton so it will boil down to whether he wants to be a professional footballer who plays week in week out, or if he values the financial security Everton will offer him for at least another season and put that ahead of first team football.