Pirlo was injured during the first half of Milan's 1-0 home defeat by AS Roma on Saturday, the Rossoneri struggling to control the game and conceding the goal after he went off.
Serie A now has a mid-season break until January 6 and Pirlo will travel with the squad for warm weather training in Dubai later this month.
Milan are trying to sign Antonio Cassano from Sampdoria in time for their next game at Cagliari and the loss of Pirlo's creativity makes the move even more of a priority.
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