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Sutton expressed concerns following the club's poor start in the Europa League, where they have lost all three of their opening matches.
Sutton said (RecordSport): "What decent player is going to want to come to a club that is seemingly happy to be European also-rans at best and whipping boys at worst? Seriously.
"There has probably been high-fiving and backslapping in the Celtic boardroom about getting the likes of Camilo Duran and Kasper Hogh.
"But those players were likely signed with the promise of competing in the Champions League – not propping up the Europa."
He also raised concerns about the club's ability to retain their current top players.
He suggested that if a significant offer were made for a star like Duran, it might be difficult to keep him if the club's ambitions remain limited.
Sutton added: "And the same goes for hanging on to their decent players. If a big bid comes in for someone like Duran, why wouldn't he jump ship if it's a club that is going places?"
Celtic are set to face Hibernian at Easter Road this weekend. They currently sit top of the Scottish Premiership.