The 36-year-old who has made a club record 828 appearances for United, suffered the injury during a collision with Villa's Steve Sidwell after 74 minutes of the Premier League match which ended in a 1-1 draw at Villa Park.
In a statement on their website the club confirmed the results of an X-ray on his right arm, saying he would definitely miss the League Cup final against Villa at Wembley Stadium on February 28.
He will also miss the Champions League match against AC Milan next week, at least six league matches and possibly the return against Milan at Old Trafford on March 10.
Giggs has featured in 21 of United's 38 matches this season.
He has collected more individual honours than any other player in the English League, with his major honours comprising 11 League titles, four FA Cups, three League Cups and two European Cups.
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