The Norwegian's agent, Tore Pedersen - who also represents Stefan Johansen - has been in lengthy negotiations with the Parkhead club.
Deila is understood to have told Stromsgodset he wants to take up the offer to become Neil Lennon's successor and is expected to be with his representatives when they travel to Glasgow this afternoon.
Today it emerged former Hoops midfielder and Hibs boss, John Collins, is being lined up as his assistant.
SportTimes broke the news yesterday that 38-year-old Deila was the man Celtic wanted to fill the position which has been vacant for 14 days.
The final obstacle to be cleared is agreeing compensation with the Norwegian champions.
Talks between the clubs have taken place, but Stromsgodset are reluctant to lose the man who, in November, led them to their first title in 43 years and who, after 11 games, has them in second spot behind Molde in the defence of their crown.
Deila signed a new contract in 2013 and it's scheduled to run until the end of 2016.
Celtic insist they have other candidates on their list of possibles.
But it is clear from the way sources at the Parkhead have been talking about Deila that the up-and-coming coach is the one they want.
After watching his side lose in extra-time to Tromsdalen last night, and exit the Norwegian Cup, Deila was reluctant to discuss the approach.
He said: "It's speculation. There's nothing concrete. It's flattering. It's a big club, and it's nice to be one of the names mentioned."
He was uncomfortable when pressed on the fact he had previously intimated his desire to move abroad.
He told interviewers: "That's also speculation, but I always said I have a dream. This isn't the right day to talk about coaching internationally, but we'll see when the time comes."
Deila is now expected to hold talks with Stromsgodset.
Having watched him develop the team since taking over his first coaching position there six years ago, and with their first appearance in the Champions League qualifying stages coming up next month, the Norwegians are unlikely to allow Celtic to recruit Deila without a fight.
Chairman Tom Saxegaard said: "We want to keep him. We really think he can get to the very top."
Chief executive, Erik Espeseth, added: "It's always fun when there is speculation about players and coaches in Stromsgodset.
"It shows we have arrived on the map of Europe."