Bryan, who can also play as a midfielder, is yet to make a first-team appearance for City.
The 19-year-old is captain of the Premier League side's development squad, helping them reach the FA Youth Cup final last season.
Walker, 20, has scored one goal in 20 first team appearances for Bolton.
Find all the latest football transfers on our dedicated page.
League One side Bury have signed versatile Manchester City centre-back Kean Bryan and Bolton midfielder Tom Walker on season-long loan deals.