With Graham Potter agreeing to take up the Chelsea job this week, Brighton are now in the market for a new manager.
Postecoglou’s work north of the border has been noticed and the kind of football Celtic have been playing has piqued the interest of several observers.
The Australian tactician is believed to be one of the managers Brighton are looking at, but Sutton stressed that he is not going to leave Celtic for a move to the south coast of England.
He feels the Bhoys are a bigger club than most of the sides in the Premier League and while he believes one day Postecoglou will leave, Sutton stressed that it will be for a club in the top six.
The former Bhoy wrote in his column for the Daily Record: “If you back yourself, you are going to be selective and not jump at the first opportunity that comes your way.
“The Premier League is regarded by many as the best league in the world.
“But Celtic are far bigger than the majority of teams who operate within England’s top flight and that is a fact.
“Like I’ve said, no disrespect to Brighton, but Celtic are miles ahead of them.
“I can’t see a coach of Postecoglou’s quality stepping down the ladder to go to England’s South Coast.
“When the day comes for Postecoglou to leave and if it is to England, it’ll be to one of the bigger clubs, a top-six club. If it’s not there, I don’t think he’ll be going anywhere.”
Sutton does not see the parallels with Brendan Rodgers as he feels the current Leicester boss’ relationship with the Celtic hierarchy was not the greatest and he had already achieved what he had to at Parkhead.
“It’s understandable some Celtic fans may be concerned about the situation given their experience with Brendan Rodgers when he walked out of the door.
“However, I believe the circumstances around that are quite different.
“Rodgers had been at Celtic for over two-and-a-half years when he moved to Leicester and, from the outside, it appeared the relationship between boss and hierarchy was not as sweet as it was in the beginning.
“You got the feeling he had taken the club as far as he could.”
Postecoglou also has Champions League football to tackle at Celtic this season.