Shaw was taken to the St Anna Ziekenhuis hospital on the outskirts of Eindhoven on Tuesday night after breaking his right leg in two places during United’s 2-1 defeat to PSV.
The England left-back broke his tibia and fibula following a challenge by PSV defender Hector Moreno in the 15th minute.
Two orthopaedic surgeons and a trauma specialist conducted an operation on Shaw’s leg on the night of the match.
United declined to comment any further but it is understood that Shaw will have a second operation at the Dutch hospital today.
Shaw is said to have suffered a “clean break”, which makes it more likely his recovery will be quicker.
The player himself has not set a target for his comeback date, but Louis van Gaal said after the match that he could return by March.
“I am not a doctor so I cannot say that, but when you have a double fracture it is I think until six months,” the United manager said.
“It is about how it develops. He will not play in the group phase I think. I hope he can play again this season.”
Former Coventry defender David Busst suffered a horrific leg break against United in 1996 after colliding with Denis Irwin and Brian McClair and retired from the game a few months later.
Busst needed 26 operations and contracted the MRSA virus.
But he is confident Shaw will make a full recovery. He said: “It wasn’t the actual fractures that stopped me from playing, it was the infections. But so long as it’s a clean break and there is no infection, there shouldn’t be a major issue for Luke.”