The former Parkhead striker believes the Northern Irishman - who has a deal until 2021 - may feel he’s reached a ceiling with the Hoops, and fears comments he made after Thursday’s Europa League defeat at the hands of Valencia have hinted at his future.
Speaking on BT Sport, Sutton said: “I think [Rodgers has] got a massive prioblem in Europe now and I think a lot of Celtic fans have cottoned onto it.
“His European record isn’t very good at all. I’m worried about what he said, comments about reaching a certain level.
“I think there are a lot of Celtic fans who are thinking that he’s concentrating on the Treble and then he will be off in the summer.”
Sutton feels the current state of Celtic’s squad - “short-term loans going back, no centre-half pairing” - means the club will be in a mess if Rodgers leaves.
He continued: “I think Brendan owes it to the Celtic fans to come out and tell everybody that he wants to stay next season.
“Because, if he does go, Celtic are in a mess in the summer... it’s going to be a massive problem. I think there’s a fair chance that he will be off in the summer.”
As well as the defence, Sutton feels his former club need to address their attacking options, insisting that “Edouard isn’t a natural number nine.”
He explained: “The truth is you can’t play at this level in Europe without a recognised centre forward and Celtic have got a problem.
“Edouard is a problem because I think he isn’t a natural number nine. He doesn’t play that role.”
Timothy Weah and Oliver Burke were brought in during the January transfer window, along with Ivorian striker Vakoun Issouf Bayo.
Leigh Griffiths is also unavailable for the foreseeable future, and Sutton feels attack is still a problem for Celtic.
Burke has also been utilised as an out-and-out striker, despite playing as a winger for previous clubs Nottingham Forest, RB Leipzig and West Brom.
Sutton said: “Burke, as hard as he tries - and he’s been very very good domestically - he doesn’t know how to move at that level, and it’s about getting up the pitch.
“There were times where he didn’t know how to show for the ball, or get on the side for the ball. And that’s been Celtic’s problem all season in Europe.”