The two clubs agreed a £2million compensation deal on Saturday following a protracted stand-off.
Mowbray looks set to be unveiled at Celtic Park in the early part of this week, and all the indications are that he will be a popular appointment.
The former Middlesbrough skipper, who invented the now famous Celtic huddle during his playing spell at Parkhead, impressed with his attractive brand of football during his first managerial job at Hibernian.
And he maintained his popularity among the Baggies faithful despite finishing bottom of the Premier League this season, one year after winning the Championship title.
Mowbray's former Celtic team-mate, Malky Mackay, believes his football philosophy makes the Englishman the ideal successor to Gordon Strachan.
Mackay, who himself looks set for a new job this week as Watford manager, told Sky Sports News: "I was asked a couple of weeks ago when I was up for Tommy Burns' testimonial day who would be my tip.
"I said then I felt Tony Mowbray's style of football would be embraced by the Celtic supporters, and they would take to him even more than the relationship they have got with him at the moment."
West Brom fans have reacted with "bitter disappointment" to the prospect of Mowbray leaving after less than three seasons at the helm.
But Alan Cleverley, secretary of the West Brom Supporters Club,
has told Celtic fans they can look forward to an exciting brand of football if Mowbray returns to Parkhead.
He said: "I think the fans took to him because he was a decent, upright bloke who gave honest answers to questions.
"The style of football he played for two-and-a-half years was always exciting.
"It was unbelievable at times how we would outplay some of the top teams, but the club's finances and the failure to buy a striker cost us dear.
"We had two appearances at Wembley, a championship win, an FA Cup semi-final and two seasons with over 100 goals.
"The brand of football this season was such that you always felt we had a chance of winning no matter who we played."
Cleverley added: "There's a strong bond between him and the fans and if he goes to Celtic I personally, and the supporters, will be bitterly disappointed.
"Most of it comes from a statement from himself at the end of the season when he asked players to show loyalty and stay for at least one more season.
"As far as I'm concerned that loyalty should count for the manager too.
"But I knew as soon as Gordon Strachan left they would come looking for him. I was just hoping they might get someone else first. I'm still hoping it will fall through."
If the deal goes through as expected, Mowbray is likely to bring along his long-time assistant, Mark Venus, and first-team coach and former Celtic midfielder Peter Grant.
And speculation has already begun as to who his first signings will be.
West Brom captain Paul Robinson has admitted he would like to follow Mowbray to Celtic Park while another left-back, out-of-contract Real Sociedad player Xabi Castillo, is apparently keen on moving to Celtic.
Midfielder Borja Valero and striker Marc-Antoine Fortune, who was on loan from Nancy, have also been linked with a move to Parkhead from Albion.