The Scotland international is out of contract at Parkhead later in the year, but as yet has not put pen to paper on an extension.
That uncertainty has led to questions over his future in Glasgow, with various reports suggesting that he is being targeted by a number of clubs this summer, including Crystal Palace and Sampdoria.
But Sutton is in little doubt that his former side should hang on to the 26-year-old, who featured in Tuesday evening’s 1-1 Champions League qualification draw against FC Midtjylland.
Writing in a column for the Daily Record, he said: “Ryan Christie doesn’t look like a man pushing to get out of Celtic
“In fact, I’d say it’s the polar opposite.
“Watching him against Midtjylland in midweek I saw a player with total commitment to the club and the ability to play a vital part in the future.
“Celtic should be going to him with a contract offer right now that he simply can’t turn down.
“It’s always been puzzling why some of the Parkhead support haven’t really taken to the playmaker.
“Fair enough, he might not always be brilliant but he never hides, he never seems to lack desire and he certainly seems to give everything.
“Christie is an international player who can cover a variety of positions, he scores goals, he creates them and he plays with a high intensity.
“Forgive me if I’m wrong but that sounds like a perfect player for Ange Postecoglou.”