The experienced duo have played important roles for the Toffees this season and their manager is now hoping to convince the board to extend to their expiring deals.
Everton manager David Moyes is keen to tie down veterans Sylvain Distin and Phil Neville to one-year contract extensions, with their current deals set to expire at the end of the season.
The two defenders have played important roles in the Toffees' encouraging start to the season and, although they are 36 and 35 years old, respectively, the Scot believes that they still have much to offer at Goodison Park.
"We are monitoring Phil's position, but, if you were asking me, I'd say: 'Let's give him another year's contract and keep it going'," Moyes told the Liverpool Echo. "I'd say the same for Sylvain Distin, who is in the same position.
"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."
Club captain Neville is currently taking his coaching badges in preparation for the conclusion of his playing days but the former England international insists that he is not ready to retire just yet.
"I want to keep playing for as long as I can - at least one more year at the top level if I can," he said. "I still have ambitions. For instance, I've always wanted to sample playing abroad, a different culture and different life-experience for all the family. Time's running out, I know, and if the gaffer comes to me in January and wants me to stay another year then the decision's made for me."