The Ivory Coast-born 20-year-old, the final signing by Moyes' predecessor Alex Ferguson before his retirement, joined the Premier League champions this week for his first training session.
United have in the past loaned out promising youngsters to other sides to gain experience and there has been speculation that Zaha, who has yet to play in the Premier League, might return for a spell at previous club Crystal Palace.
Palace boss Ian Holloway has effectively ruled that out and Moyes indicated the once-capped youngster was part of his plans.
"He will travel with us on the tour," he told talkSPORT radio before United's departure to Thailand for a tour that starts in Bangkok on Saturday.
"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."
Skipper Nemanja Vidic is a surprise absentee from the touring squad.
As anticipated, Robin van Persie and David de Gea both miss Saturday's opener in Bangkok after being given additional time off following their summer international commitments, although they are due to join up in Sydney.
And, just as Moyes predicted, injury victims Chris Smalling (foot), Ashley Young (ankle) and Nani (broken nose) were also missing due to injury.
However, Vidic's non-appearance was not expected and there was no immediate explanation from the club.
There was no hint of any problem on Friday when Vidic attended the official unveiling of Moyes.
And while his absence may be connected to the treatment received by the Serbian on the chronic knee injury that ruled him out for virtually 12 months, any fresh problem would be a concern.
In addition to Van Persie and De Gea, Shinji Kagawa will also be a late arrival, linking up with his team-mates in Japan on July 21 after a rest following his country's Confederations Cup campaign.
Moyes reiterated that England striker Wayne Rooney was not for sale and would not be leaving United.
"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," said the Scot, who was manager of Everton when Rooney began his career at that club.
"He is training great. They are telling me it is the best shape they have seen him in for a long time... I am going to try and get the best Wayne Rooney I possibly can."
Moyes said United were also working on bringing in new players.
"Don't be kidded because there is a bit of a lack of activity," the manager said. "Take it from me, we are working hard behind the scenes to try and improve the squad.
"I won't talk about players at other clubs, I don't think it is right, but I can tell you we are out there trying to do some business."