The 38-year-old Norwegian has come from left field to emerge as a serious candidate to fill the vacancy left by Neil Lennon 13 days ago.
Senior sources at Parkhead have confirmed Deila is one of the men on the list they have compiled, and that he is hugely admired for the work he has done with Stromsgodset, who he has just led to their first Norwegian championship for 42 years.
However, the Hoops face a battle to prise the qualified teacher away from his current position.
Stromsgodset insist they have not had any contact from Celtic.
When asked today about Deila moving to Parkhead, their Sports Co-ordinator, Geir Amundsen, said: "I have absolutely no comment to make on this because we have not heard from Celtic."
He did confirm, however, that the much-sought-after Deila is still contracted to the club where he stepped up from player to become manager in 2008.
Amundsen said: "He is under contract with us for, I believe, two or three years."
Deila has built up a reputation for producing young, exciting teams, and developing players such as Stefan Johansen, who joined Celtic from the Norwegian champions in January.
He favours an attacking system based on hard work, speed and organisation.
It has already paid big dividends with Stromsgodset - who were nearly bankrupt in 2006 - winning the Norwegian Cup in 2010 under their progressive coach and their Director of Football, Jostein Flo.
They took Stromsgodset to runners-up spot in the championship in 2012, and went one better last season, beating much-better resourced clubs such as Molde and Rosenborg to lift the trophy in December.
Deila's style has seen him compared favourably to Borussia Dortmund coach, Juergen Klopp, who he has gone to Germany to meet and to study.
He has also visited Brendan Rodgers at Liverpool in his quest to increase his coaching knowledge.
With his stock so high, it would be considered a major coup if Celtic succeed in landing Deila.
The Hoops would prefer to get any deal done as quickly as possible in case any other clubs who have been monitoring come in with a counter-proposal.
But Celtic will also continue to consider others already on their list of candidates, including Steve Clarke, Malky Mackay, Owen Coyle and Michael Laudrup.
The indications are that hopes of persuading Henrik Larsson to return from Sweden to the club where he made his name are now receding.