Mackay-Steven joined Celtic from Scottish Premiership rivals Dundee United in February 2015.
The 25-year-old winger played his role in the Bhoys’ league success last season, but he has not been playing well so far this campaign.
Former Celtic striker Sutton has questioned the Scotland international’s mentality.
“If you take the Dundee United lads - Gary Mackay-Steven and Stuart Armstrong - they kicked the team on last season,” the Englishman told The Scottish Sun.
“Armstrong has had injuries but you would have to question Mackay-Steven’s mentality. He seems like a nice guy but what’s gone on there?”
According to a recent report in The Glasgow Evening Times, Mackay-Steven could be loaned out in the January transfer window.
The former Airdrie United winger needs playing time, but he is not getting that at Celtic at the moment. Perhaps it would be best for the development of the winger if he went out on loan to a Scottish Premiership club for the rest of the season.
Celtic are at the top of the Scottish Premiership table at the moment with 46 points from 20 matches. Ronny Deila’s side are three points clear of second-placed Aberdeen and also have a game in hand.