He told BBC Radio Lincolnshire: "Making sure they are solid and staying in the Championship would be a good season.
"Fans do have wild expectations but I'm sure the management and board will be thinking about stabilising."
Sutton's current team face his old club in a friendly on Wednesday with the ex-Blackburn and Celtic frontman warning City to keep their feet firmly on the ground.
"If they could go up then fantastic, but it's not easy and they won't find it as easy as they did last season," he added.
"They have players who can cope at Championship level but it will be tough for them next season.
People from other parts of the country probably don't appreciate it but when you come here you really do
Norwich boss Paul Lambert on the club's supporters
"I think they'll do well, I just think they've got to be careful in not getting too far ahead of themselves and the fans thinking they can win the league outright."
The level of expectation among supporters has been displayed by the club reaching its season ticket threshold.
City have sold a club record of 20,550 season tickets and will now stop sales, with only casual tickets available for home games.
Boss Paul Lambert told the club website: "Our fan base is phenomenal, it really is.
"People from other parts of the country probably don't appreciate it but when you come here you really do.
"Now we must aim to reward their loyalty by performing well on the pitch."
Source: BBC Sport
Source: BBC Sport