Shaw was brought off in United's defeat to PSV in the Champions League.
It was suspected he had broken his leg and the club have now confirmed it.
In an announcement on their official Twitter account, United said: "Our thoughts go out to Luke Shaw, who has suffered a double fracture of the leg. He will have an operation on his return to Manchester."
The 20-year-old went down under a challenge from Hector Moreno early in the first half of United's 2-1 defeat to PSV.
Medical staff rushed onto the pitch and the defender received on-pitch treatment for several minutes.
Our thoughts go out to @LukeShaw23, who has suffered a double fracture of the leg. He will have an operation on his return to Manchester.
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