David Moyes and Roy Keane, who both played for Celtic themselves, are among the bookies' favourites to replace the outgoing Ronny Deila this summer.
But according to Sutton, it is unlikely that the pair will want to work under the current framework installed by Celtic chief executive Peter Lawwell, who's been accused of controlling transfers and behaving like a director of football.
"The issue Celtic have is Peter Lawwell and whether he is going to relinquish his grip on a new manager when he comes in," the 43-year-old told the Herald.
"Is a David Moyes going to want to come in and work under Peter Lawwell in the existing framework? I would argue that he wouldn’t."
He added: "Roy Keane has said he wants to manage again and that is no secret.
"But if Keane turned it down once because of that situation, because he did not want to be told what to do from above, then it would shock me if he took the job under the same circumstances."
As reported by the Mail on Friday, John Hartson echoed his former teammate's opinion on Lawwell and suggested the club must appoint a strong personality to stand up to him.
Lawwell, though, has insisted that such claims are not true and added that the new manager will have "100 per cent control" of transfers.
Celtic's Champions League qualifying campaign kicks off in ten weeks time, and Sutton also expressed fears that the club's failure to act fast to Deila's departure decision could leave them "vulnerable", considering the "major surgery" required in the transfer market.
The Bhoys have failed to qualify for the group stages of Europe's premier club competition since 2013-14.