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 on the matter, but it is understood that Shaw will have a second operation at the Dutch hospital as part of his recovery on Friday.
Shaw will remain in Eindhoven until United's medical staff are happy for him to come home.
United executive vice-chairman Ed Woodward visited Shaw the morning after the match and the player's parents are understood to be with him now.
Shaw is said to have suffered a "clean break" of the right leg, 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
You can never say that in advance.
''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 at Old Trafford 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 told The Sun: ''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.''
Source : PA
Source: PA