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.
Giggs has featured in 21 of United's 38 matches this season.
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