The 33-year-old's current deal is set to expire in the summer but he is yet to agree terms on an extension, with the fine form of Seamus Coleman seeing him limited to just six appearances in 2013-14.
Manager Roberto Martinez, though, stresses the importance of retaining a player such as Hibbert, who has made 288 appearances for the Toffees and was afforded a testimonial last summer.
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"Maybe, in a couple of key moments, he's picked up a couple of injuries that have stopped him playing more but I do value his contribution at the club and his experience and I would love to keep a character like Tony Hibbert, who means so much to us.
"He's played such a significant role and sometimes the fans don't see that. Sometimes you can get misled because you don't see a player playing week in and week out and you can feel that they don't have a major role in the football club.
"Tony has had a few niggles this season but, in terms of his day-to-day work, he's been magnificent and I'd go as far as saying he's been paramount in keeping really strong standards in our training sessions, in making sure that the group was ready, helping out the youngsters and every time he has played he has performed well."