The Sun says Moyes has even told the £15million signing from Crystal Palace to get changed next to Giggs in the dressing room.
New Old Trafford boss Moyes hopes Zaha will be able to learn what it takes to be a top pro like the evergreen Welsh man, who is still playing at the highest level aged 39.
Zaha, 20, has impressed Moyes in training and during pre-season games. But the Scot intends to make him much more disciplined in his approach.
The winger will not be loaned out in this transfer window and Moyes will monitor his progress before deciding if a January loan move could aid his development.
But for the moment United are hoping that Giggs' example in the dressing room will have a positive impact on the Three Lions ace.