The 31-year-old midfielder revealed he has held talks about the possibility as he prepares to make his 400th Hoops appearance against St Johnstone at Parkhead tonight.
Brown is in his 10th season at the champions after Gordon Strachan paid £4.4million to Hibs for the dynamic performer – which is still a record deal between two Scottish clubs.
He is under contract until the summer of 2018 and has held talks with chief executive Peter Lawwell over an event to mark his service.
Brown admitted: “We have been speaking, me and Peter. We have been having a chat about it, so it could happen.
“But I’m in no rush whatsoever, here’s hoping I have another good few years left here. If they want to offer me another four-year deal, I would be happy to accept.
“I have still got the legs and the love for the game, as well, and that’s the main one. I feel as good as I have ever felt in a Celtic jersey.”
Brown is closing in on his sixth consecutive title as the Parkhead onfield leader and a seventh championship winners’ medal overall.
He could join a glittering array of Hoops heroes who have been honoured by the club in such a way.
Lisbon legend Billy McNeill and iconic manager Jock Stein both received money-spinning benefit games.
Jimmy Johnstone and Bobby Lennox shared their big night against Manchester United in 1976 and others including Tommy Burns, Paul McStay, Roy Aitken, Davie Provan, Danny McGrain, Pat Bonner, Peter Grant, Tom Boyd and Jackie McNamara have also received windfalls from the club.