Wan-Bissaka picked up the injury in Saturday's 3-1 home defeat to Brighton and Hove Albion in the Premier League.
The 25-year-old was only an 85th minute substitute.
Wan-Bissaka has made five appearances in the Premier League this season but will now be missing for a number of games starting with Wednesday's Champions League encounter at Bayern Munich.
“Aaron Wan-Bissaka is set for a period on the sidelines after sustaining an injury during the closing stages of Saturday’s game against Brighton and Hove Albion,” read a United statement.
“Further assessment will be needed to determine how long Wan-Bissaka will be out for, but initial indications suggest it will be several weeks.”
Manchester United are now facing a crises in their full back positions, with Luke Shaw and Tyrell Malacia also out injured.
Loan signing from Tottenham Hotspurs, Sergio Reguillon and Diogo Dalot are the only full back options available to manager Erik Ten Hag.
