The 28 year-old, who joined Celtic’s youth set-up aged 13, has three years remaining on his current deal but has expressed an eagerness to extend his stay beyond 2022.
Manager Neil Lennon, who handed Forrest his first-team debut back in 2010, described the Scotland international as an “outstanding player” who could play anywhere.
Lennon said: “He’s a top, top player. He’s been brilliant. From the day I first saw him until now, over that 10-year period he’s been magnificent for the club. He’s achieved a lot but there’s still that hunger in him.
“He wants to extend his contract as well which is great. Rather than talking about people leaving the club we’ve got one there who wants to stay. So that’s one we’ll look at. He’s still got three years left on his deal but we’ve already talked about an extension.
“I think he got a bit maligned a little bit for a spell or two but over the last few seasons he’s really matured into a top quality player. He could go and play anywhere but I’m glad we have him. He’s a brilliant player and I’ve never hidden my admiration for him.
“He’s player of the year, a treble-treble winner, he’s played in the Champions League, and was international player of the year for Scotland. He’s been doing it year in, year out.”