Former Foxes boss Claude Puel was sacked on Sunday evening following a run of poor results for Leicester.
And despite Leicester managing to obtain wins over Chelsea and Manchester City this season, as well as a draw at Anfield, Puel couldn’t keep his job.
And almost immediately, one man was linked with the now vacant post: Brendan Rodgers.
Rodgers has been a success in Glasgow where he’s been in charge of Celtic for the last two-and-a-half years.
He’s won two Scottish Premierships and looks on course to win a third one this season.
And Sutton believes that Sutton should stay with Celtic, despite interest from a Premier League club.
Mark Champman queried Sutton on what he thought of the possible appointment.
He asked: ”If Brendan Rodgers asked you should he take the #LCFC job, what would you tell him?”
“I’d say don’t downgrade yourself and stay at Celtic,” Sutton replied.
“You get European football at Celtic, and you won’t get that with Leicester.
“I don’t want to see him leave Celtic but I think it will happen.
“He could have dismissed it yesterday and he chose not to dismiss it before the Motherwell game on Sunday.
“If somebody asked Brendan if he was linked with the Bury job and there was any interest, he would have said ‘don’t be daft!”
And on Puel’s sacking, Sutton showed empathy to the situation, describing it as “harsh”.
He said: “This is a very harsh sacking in terms of what he was building there, in terms of young players coming through and in terms of net spend, he was doing an okay job.”