The outspoken pundit has been discussing Celtic's transfer policy.
Sutton is quoted as saying by the Mail: "Would David Moyes want the job? Would someone of his ilk want it if he doesn't have control of what they want to do? That's a problem and it makes it not easy to choose who the next manager could be.
"I don't know what the transfer policy is at Celtic but we all have our ideas about what's going on. I spoke to Ronny earlier in the season and he said he gets a say and you have to take him at his word. But it's clearly a problem.
'"The signings of Scott Allan and Ryan Christie were strange. They must regret moving to Celtic. They'd be better off playing first-team football at their old clubs."
A host of names have been linked with the Celtic job after Deila's end-of-season departure date was announced.
Moyes is just one, with Neil Lennon, Brendan Rodgers, Ryan Giggs and more all linked at one point or another.
This Celtic appointment is far more important than the last one. However much fans may not like it or want to acknowledge it, the Bhoys will have proper rivals once again next season.
Rangers have already beaten the Hoops this term and pretending they aren't a threat next campaign isn't going to help anyone. Celtic need a manager who's aware of the rivalry - and ready to beat the Gers.