The Foxes – who axed Claude Puel over the weekend – unveiled the 46-year-old earlier today.
Sutton, however, took a sly dig at the experienced chief.
“Good luck to Brendan Rodgers in his new dream job…” the former striker turned pundit posted on Twitter.
Rodgers has signed a deal that will keep him at the King Power Stadium until 2022.
His successor at Celtic Park will be Neil Lennon who previously managed the club from 2010-2014.
Sutton wished Lennon luck, but couldn’t resist taking another pot shot at Rodgers.
He added: “Good luck to Neil Lennon at Celtic… not an easy hand to take on… tough gig with injuries to key players… but the positive is he will see the job through…”
Lennon will remain in charge until the end of the season.
“There’s a chasm to fill with Brendan leaving,” Lennon told Celtic TV.
“He’s done a remarkable job, but I’m here for the club.
“The club comes first as do the players. I want to bring as much success during my time here as I possibly can.
“All I want to do is carry that on – it’s not broken and I’m not coming in and looking to revolutionise anything.”
Celtic chief executive Peter Lawwell reckons the club legend will hit the ground running.
“In bringing Neil back until the end of the season, we have appointed a man who knows what it means to be Celtic manager and someone who knows how to win,” Lawwell explained.
“We know he has the commitment and desire needed to face the challenges ahead.
“We are delighted to also promote John Kennedy to assistant manager and bring Damien Duff into our first-team management group.
“This will bring vital continuity and allow us to build on Brendan’s legacy.”
Lennon previously led Celtic to three Scottish Premiership title.
His first game back in the hot seat will be against Hearts on Wednesday.