The pair's current deals both expire at the end of the season and manager David Moyes is keen to extend their stay at Goodison Park.
Neville, who turns 36 in January, has played in 10 of the club's 11 Premier League matches this year, while Disin, who is 35 in December, has appeared in nine.
Moyes is eager for the duo to use their experience to help guide the rest of the team.
"We are monitoring Phil's position, but if you were asking for me, I'd say let's give him another year's contract and keep it going," said Moyes. "I'd say the same for Sylvain Distin, who is in the same position.
"With all due respect, when you get to their age, they understand most things are on one-year deals or rolling contracts. Both of them have been excellent and I'd have no qualms recommending to the board that we give them another years contract.
"Phil and Sylvain are both in the same boat. I've not really sat and had a word with the chairman yet, we'll probably do that after the New Year, and see if we are ready to give them a new contract."
Everton travel to Reading in the Premier League on Saturday.