Zaha only began pre-season training on Monday after being given extra time off following England's European Under-21 Championship campaign and Moyes wanted to see the former Crystal Palace man before deciding whether to pick him.
However, Zaha has apparently done enough to be included in the United party when it leaves for Bangkok later today.
"He will travel with us on the tour," Moyes told talkSPORT.
"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."
Wayne Rooney will also be there, although it appears there have been no further developments on his future, with Moyes virtually repeating what he said at Old Trafford on Friday, when he insisted the 27-year-old would not be sold.
"The club have already stated Wayne is not for sale," the Scot said.
"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. He came to my house, I have been to his recently to have a chat. Also he is here every day.
"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."
Moyes also confirmed work was continuing at beefing up his squad.
An increased offer for Everton full-back Leighton Baines is anticipated, while United remain interested in Barcelona midfielder Thiago Alcantara.
"Don't be kidded by the lack of activity," Moyes said.
"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.
"We are also very interested in developing our own players and bringing in potential players for the future. We are working on that just now."
United's tour takes them to Thailand, Australia, Japan and Hong Kong before they return home for a further friendly in Sweden and then Rio Ferdinand's testimonial against Sevilla at Old Trafford on August 9 and the Community Shield two days later.
However, it is that first Premier League game at Swansea - followed by the high-profile encounters with Chelsea, Liverpool and Manchester City - that are the main focus ahead of what is one of the most eagerly awaited Premier League campaigns.
"It is the first time there has been a real major change in the Premier League with the managers at the top end changing round," Moyes said.
"Jose (Mourinho) is coming back. He will be terrific at Chelsea. He has been a success wherever he has gone. Arsenal and also Manchester City, who have a new manager, and Tottenham are trying to challenge.
"I really do see an exciting Premier League. Lots of people will be hoping it is not the same as last year.
"The only people who do are me and the Manchester United players. We are hoping that it is the same as the way it finished last year."