A former Celtic, Norwich, Reading and Bournemouth midfielder, Grant has been named as replacement for Roy Aitken who left Blues to join David O'Leary at United Arab Emirates outfit Al Ahli last week.
Grant began his coaching with Bournemouth before leaving the Cherries in 2004 to become coach at West Ham.
He spent two years at Upton Park before joining Norwich, who he managed for a 12 month period, before teaming up as number two to Tony Mowbray at West Bromwich Albion.
Grant moved to Celtic when Mowbray was appointed manager at Parkhead last summer.