The 19-year-old joined Old Trafford in a £30 million deal from Southampton this summer but Van Gaal declared that the left-back "is not very fit" during his side's tour of the United States in July.
Shaw was forced to train on his own in order to improve his fitness before he picked up a hamstring injury which ruled him out of the start of the Premier League season and Hodgson has revealed that the left-back's fitness levels were questioned while England were in Brazil, though noting that the youngster's body has yet to fully develop.
"I must say, watching him play in the matches for Southampton, he didn't seem to have too many problems coping with the physical side of the game but that doesn't mean to say he can't be fitter and if Louis thinks that he can get him fitter then I'm sure he will certainly have our approval," Hodgson told reporters.
"I think when [Van Gaal] talks to [former Saints boss Mauricio] Pochettino then Pochettino will say to him: 'What you were saying to Luke Shaw and what you've been saying to him is what I was actually saying to him.'
"He might have had it mentioned to him a couple of times during the World Cup when we were together by people like myself and [England head physio] Gary Lewin.
"So I think he's been aware of it, Luke, and he will get fitter and stronger, not least of all because he's still a very young man and his body has not yet reached its proper proportions for the level he wants to play at."