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.
And on Tuesday, Rodgers verbally agreed to join Leicester, with Celtic releasing a statement explaining that they'd let the former Liverpool manager speak to the foxes.
Mark Champman queried Sutton on what he thought of the possible appointment.
Chris Sutton believes Brendan Rodgers should stay at Celtic (Image: GETTY/BBC )
Brendan Rodgers has won two Scottish Premierships (Image: GETTY )
Puel was sacked on Sunday by Leicester (Image: GETTY )
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Brendan Rodgers to Leicester: He'll be OFF says Chris Sutton
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.
Brendan Rodgers could take over at the King Power (Image: GETTY )
“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.”
Neil Lennon will reportedly take over on an interim basis at Celtic - the club he previously won three Scottish leagues with.