Alex McLeish has been looking for a new right-hand man since the departure of Roy Aitken last week.
And despite interest in former West Ham and Chelsea coach Steve Clarke, they have opted to bring in former Norwich manager Grant.
"He is a competitive man who has been brought up in a winner's environment due to the traditions of Celtic. He has successfully made the transition into becoming a coach and my diligence shows me he is the right man for the job.
"His personality is akin to that of the man we have just lost, Roy Aitken, and that will be a great asset to us.
He left his player-coach post with the Cherries in 2004 to become Alan Pardew's No 2 at West Ham, the pair guiding the Hammers to the FA Cup final in 2006.
A 12-month spell as Norwich boss followed in 2006/7 before he became first-team coach at West Brom in summer 2008 under the management of Tony Mowbray.
He followed his former Celtic team-mate back to the Glasgow giants last summer for the ill-fated stint back at Parkhead which ended in their dismissals in March this year.