The Hoops potentially have 10 more games left this term.
The Norwegian has come in for criticism this term for seemingly failing to take the Bhoys forward. With their time in Europe proving to be a failure, and their League Cup run ending in a surprise defeat in the semi-final, the club go into the final part of the season having failed to secure a trophy thus far, and knowing that they can only match last term's haul of two pieces of silverware.
And Sutton suggested that even if Deila is able to oversee a flawless run that leads them to both the Scottish Premiership title and Scottish Cup crown, the writing should be on the wall already for his time at Parkhead.
"They need to glitter through the final eight games of the Premiership in style," he told the Record.
"While doing so, Rangers have to be dispatched in the manner a Championship side should be dispatched in the upcoming Scottish Cup semi-final. The trophy needs to be lifted in May with a handsome performance.
"It wouldn't make any difference to me or the many thousands of Celtic supporters who have decided enough is enough already. But it's not our opinions which count in the corridors of power and that is what the manager will be clinging to in the coming weeks.
"I'd say Deila conjuring 10 stunning displays is as likely as me starting up front alongside Leigh Griffiths today. But at least he might have some hope of keeping his job if he does so."
With Rangers out of the equation in the top flight, it could be argued that their performance in Europe should be considered a key factor in whether Deila is adjudged to have been a success at the club as they will be expected to win every domestic honour.
So with the side failing to make any impact once again, and also failing to dominant domestically, fans would be forgiven for thinking that Deila should be doing much better if he wishes to stay in the job for several years to come.
On the other hand however, he has given a handful of younger players a chance, perhaps in preparation for when Rangers return to the top flight - which appears to almost definitely be next season - so arguably deserves more time to allow the prospects to develop.
But if he is not able to lead Celtic on a perfect run until the end of the campaign, the calls for his departure will surely become too loud for the club to ignore.