The 38-year-old Norwegian seems sure to be officially unveiled as the successor to Neil Lennon at a press conference at Celtic Park, although there were conflicting reports from Norway as to whether a deal had been completed or was imminent.
Celtic and Stromsgodset were involved in protracted negotiations, over a compensation package, as Deila was under contract to the Norwegian champions to the end of 2016.
The man who, in November, led the tiny, provincial club to their first championship for 42 years, should begin working in his new job immediately.
The Celtic players are scheduled to return for pre-season training on June 24, and will head to their training camp in Austria four days later.
After a handful of already-arranged friendlies, their first competitive game will be the first leg of the second qualifying round for the Champions League, which takes place on July 15 or 16.
Ironically, Stromsgodset will also enter the competition at this stage, and, as an unseeded side, could even be drawn against Celtic.
Deila's first move will be to decide who he wants as his assistant as Lennon's team of Johan Mjallby and Garry Parker have also left the club and coach, Danny McGrain, is recovering from a heart attack.
John Collins is understood to be the man under consideration, and the former Celtic player and manager of Hibernian could provide Deila with a solid knowledge base of the football scene in this country, as well as a list of players who might be of interest to the new manager.
Deila already knows all about Stefan Johansen, who he sold to Celtic in January and who has proved to be an impressive acquisition to the Hoops side which just won its third title in a row.
Deila has a reputation for identifying young talent and developing players, which fits perfectly with the business plan the Parkhead club have been following for the past few years.
His teams play fast, attacking football, and this is one of the main reasons Celtic decided to look outwith the list of high-profile candidates who were believed to be leading the race to take over from Lennon.
Henrik Larsson and Roy Keane ruled themselves out of contention, opening the way for Deila to emerge from left field and pip other very strong candidates, including Malky Mackay, Owen Coyle, Steve Clarke and Michael Laudrup, to the biggest job - and challenge - in Scottish football.