The Greek stopper was brought in by the Hoops to replace loanee Fraser Forster, who returned to Southampton at the end of last season after a hugely successful spell in Glasgow.
Since then, the 26-year-old has struggled to produce his best work at Parkhead, and was even displaced in the starting XI by understudy Scott Bain for a time.
Barkas was back in the side for the 4-2 defeat to AC Milan in midweek, however, but failed to impress in another subpar showing.
Reflecting on the keeper’s performance in a column for the Daily Record, former Celtic hitman Sutton was quick to criticise the man between the sticks.
He said: “Celtic did have some nice passages of play in the game. Tom Rogic has wonderful feet and scored a sublime goal, Odsonne Edouard’s finish was delightful. So was Christie’s pass to set him up and he was unlucky with a brilliant free-kick at 3-2 which was saved superbly by Milan keeper Gianluigi Donnarumma.
“Which brings me back to Barkas. While the AC No.1 was flying across his goal to deny Christie, the Greek was standing as you would in bus shelter waiting for the No.63 as Milan’s free-kick went past him.
"It was criminal. I’m not a goalkeeper, but I do know that you organise a wall and when you do, that covers one side of the goal and you cover the other. You do not get beaten on your side.
“Barkas built a wall and then hid behind it. It was appalling stuff. He just doesn’t make saves. If he has done since he’s signed, I’ve missed them. Not much return on £5 million there and Scott Bain isn’t the answer.
“Missing out on Fraser Forster has been key to the season. The big Geordie came up with the big moments. He made the vital stops. That gave the backline a confidence which they don’t have this season. It’s no coincidence the backline is brittle with no reassurance at their backs.
“Celtic made the last 32 of the Europa League last season when Forster was making outrageous saves against Lazio. Without him, they are out of the competition.
“Forster single-handedly won the Betfred Cup Final for the first trophy last season with his saves against Rangers at Hampden. Without him, they were dumped out by County and the silverware is gone.
“Those events are no coincidence. I’ve heard people say that Celtic might be trying to get him back, but how is that going to happen? He’s warming a bench at Southampton, yet you could argue he’s closer to the first-team now than he was in June.
“If Celtic couldn’t make a deal happen then, how can they now? Especially given £5 million was frittered away on his replacement.”
Barkas has conceded 17 goals in 16 matches since joining the Hoops in July, keeping eight clean sheets during the spell.