The Northern Irishman was appointed by the Scottish Premiership side on Friday.
The former Liverpool boss has been named as the Hoops' new manager and will be hoping to lead them to their sixth Scottish Premiership title in a row, as well as taking them on the kind of European run that Ronny Deila could never quite put together during his tenure.
But Sutton suggested that the Bhoys squad needs big changes if they hope to make any kind of impression on the continent.
"It's not going to be easy as Rodgers now has a huge job on his hands," he told the Record.
"No one at Celtic should be kidding themselves. This is not a squad that needs a few tweaks here or there. It needs an overhaul.
"A new manager might improve the tactics and get more out of players. But when it comes to crossing the white line there is only so much a manager can do."
While Sutton is extremely pessimistic about the squad's current chances of competing at the highest level, many of the players that are currently in the side have previously shown that they have the ability to do much more than they have this past term.
As well as Kieran Tierney and Leigh Griffiths, who have been exceptional this past year, the likes of Charlie Mulgrew, Scott Brown and Kris Commons have either struggled for game-time or have spent large parts of the campaign out injured.
But if those three can rediscover their strongest form, then Rodgers will arguably have a solid spine to build around, especially with players like Erik Sviatchenko, Nir Bitton and Patrick Roberts all hinting that they could be key men for the team going forward.
So while Sutton is extremely worried, and certainly the lack of depth within the squad should be a concern, Rodgers may well be confident that there is already potential within the current side to suggest that with a couple of signings, the team could cause some surprises in Europe next season.