The centre-back, who joined the Championship club on transfer deadline day, suffered the problem during Saturday's 3-3 draw with Derby County.
"We're devastated for him. He's a terrific kid and we were absolutely delighted to get someone of his quality. Manchester United rate him so highly and he's been a brilliant lad," Birmingham boss Lee Clark told the club's official website.
It is not yet known how long the 21-year-old will be on the sidelines.