West Brom have given Celtic permission to talk to manager Tony Mowbray about the Parkhead vacancy after agreeing a £2m compensation package in principle.
Celtic will also pay an additional compensation figure for assistant Mark Venus and first-team coach Peter Grant.
It ends several days of stand-off since West Brom made their demands clear and chairman Jeremy Peace indicated he was not prepared to back down.
But West Brom announced on Saturday that a breakthrough had been made.
Peace told his club website: "We do not want to lose Tony because he has been an integral part of a long-term project at the club which the vast majority of our fans have bought into.
"But Tony expressed his desire to talk to Celtic and, now that the compensation figure in his contract has been met to our satisfaction, that process can take place.
"If Tony decides to join Celtic, he would leave with our good wishes and we would then immediately put into action the task of appointing a new manager who can take the club forward."
These are complex but important negotiations and we can assure our supporters that we continue to aim to get the right man for the job in the right way
Celtic chairman John Reid
Mowbray had been one of three managers targeted by Celtic along with Burnley's Owen Coyle and Swansea's Roberto Martinez.
Burnley's compensation demand appears to have been what thwarted Celtic in their desire to talk officially with Coyle.
Celtic then turned their attention to Mowbray after Swansea's Roberto Martinez was approached by Wigan.
John Reid, Celtic's chairman, confirmed that they would now be speaking to Mowbray.
"These are complex but important negotiations and we can assure our supporters that we continue to aim to get the right man for the job in the right way," he told Celtic's website.
Mowbray played for Celtic between 1991 and 1995, straddled by spells with Middlesbrough and Ipswich, with whom he started his coaching career.
He had a brief spell as caretaker after the sacking of present Scotland boss George Burley before becoming a manager in his own right at Hibernian.
Mowbray led the Edinburgh club to two top-four finishes in 2005 and 2006, after which he left for West Brom.
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He led the Baggies to the Championship play-offs, losing to Derby, but the following season they won the title and promotion to the Premier League.
However, they only lasted a season in England's top tier and were relegated despite receiving plaudits for their attacking style of football.
Mowbray's appointment will also result in a return to Celtic Park for Grant, who spent 15 years with the Glasgow club.
Grant, who began his coaching career as assistant boss at West Ham, spent a year as manager of Norwich before joining Mowbray at West Brom in 2008.
Tony Mowbray's pending appointment would be welcomed by Celtic fans, according to Peter Rafferty, secretary of the Association of Celtic Supporters' Clubs.
"He has served a decent apprenticeship in Scotland and England," he said. "He is obviously aware of Celtic and he will be well received by the fans.
"Peter Grant will be coming with him and nobody is a bigger Celtic man.
"Even at West Brom Tony was committed to playing good football and the fans still supported him - and that says a lot when he got the team relegated."