The two clubs have agreed a £2million compensation package for Mowbray who seems set to take on the role vacated at Parkhead by Gordon Strachan last month.
Celtic will also pay an additional compensation figure for the services of Albion assistant manager Mark Venus and first-team coach Peter Grant.
West Brom were relegated to the Championship in the summer after finishing bottom of the Premier League.
Baggies chairman Jeremy Peace said: "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."