The defender, who can still play as a winger on occasion, could become the club's oldest outfield player.
Young is currently 38 and is now the same age as David Moyes was when the latter took over as Everton manager many years ago.
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Young told talkSPORT: My contract runs out at the end of June, but I've spoken to the manager and we had a quick conversation about staying for another season. It's just now down to sorting out the final details to sort out for the next season."
On staying, Young said: Yeah I do. I'd love to play as long as possible, whether that's going to be a one-year deal or two-year deal.
Whatever everyone offered me, I'm always happy to sign a new contract. I enjoyed my time this season, being there with the players and staff, it's fantastic, a massive club and I'd love to go back and play another season there."