Zaha, who signed in January before returning to Crystal Palace on loan, reported for pre-season training on Monday after he was given extra time off in the wake of England Under-21s' European U21 Championship campaign.
And Moyes wants to see more of the winger as he continues planning for his first season as United manager.
"He will travel with us on the tour," Moyes told talkSPORT ahead of Wednesday's flight to Bangkok, where the tour of the Far East and Australia begins.
"He has only been with us for two days but I will take him away, see how he goes and have a look.
"This club has a great record of putting young players in the team. Hopefully Wilfried can be one of them."
Zaha will be joined on tour by Wayne Rooney, and Moyes has again ruled out any possibility of the striker leaving this summer.
He said: "The club have already stated Wayne is not for sale. I have met Wayne two, three, four times. A lot has been said about myself and Wayne over the years but we have a really good relationship.
"I will say he is training great. They are telling me it is the best shape they have seen him in for a long time.
"From that point of view, I am taking it as a positive and I am going to try and get the best Wayne Rooney I possibly can."
There was no further news on possible moves for Everton left-back Leighton Baines and Barcelona midfielder Thiago Alcantara, but Moyes did say: "Don't be kidded by the lack of activity.
"Take it from me, we are working hard behind the scenes to try and improve the squad. The big thing is that Manchester United will always be in for the best players in the world.
United kick off their tour against a Singha All Star XI in the Thai capital on Saturday.