The Norwegian’s position at the Scottish champions is under renewed scrutiny following the latest and arguably most damaging setback of his tenure.
Celtic supporters rounded volubly on Deila at Hampden as he suffered his third semi-final defeat in four attempts at the national stadium since his appointment in June 2014.
While Celtic remain firm favourites to retain the Scottish Premiership title this season – they are eight points clear of closest challengers Aberdeen with just the five post-split fixtures remaining – that may no longer be good enough to save Deila’s job beyond the summer.
Dermot Desmond, Celtic’s major shareholder, was in attendance at Hampden on a rare trip to Glasgow on Sunday and Deila’s future would inevitably have been the subject of discussion with his fellow directors and chief executive Peter Lawwell.
But Deila, who admitted Celtic’s performance as they lost out to Rangers on penalties after a 2-2 draw was not good enough, tried to swat away questions about his own position at the post-match media conference.
“That’s for the future,” said Deila. “The only thought I have now is to get the league title back to Celtic Park. After today, it’s hard to talk about those things. There is a lot of emotion. But with bad results, you get these questions.
“Today it’s the ups and downs of football. It’s been a rollercoaster season with no real momentum and some setbacks.
“Again that’s something we have to work through. We can’t expect pity and I can’t either. We have to go into this ourselves and turn it around.
“Today I am very disappointed. I’m not thinking about that [my future] at all. I know why the question is coming but again we have to just lift ourselves after this disappointment and get the league over the line.
“I’m not very satisfied with our performance today. We could have been better. We didn’t press well enough in the first half, we were not intense enough in that we could have intercepted more balls than we did. We need to be quicker and calmer on the ball.
“The positive thing was that we came back twice from a goal behind and created some chances, but overall it’s not the best day we’ve had.”
There was an air of defiance in Deila’s response to the question of whether Rangers can mount a serious challenge for the title next season.
“We are going to be ready when they come up,” he said. “Now we are top of the league and we are going to fight to get that title to Celtic Park.
“We will see next year what they have to offer.
“It is a totally different thing to do something over 38 games than to do it over one.”