The Brazilian suffered the injury at Carrington, United's training complex, and his position for Sunday's match at West Brom was filled by Chris Smalling.
In a statement on their website, United said: "Rafael's twin Fabio is another option in the position so Sir Alex still has plenty of cover, even though the absence of the highly-rated 21-year-old is a blow."
Patrice Evra, Javier Hernandez, Antonio Valencia and Michael Owen also missed the game at the Hawthorns due to injury.
Source: PA
Source: PA