The 18-year-old's present deal expires at the end of the season and United officials have described ongoing negotiations as "complicated".
Memories are still fresh over the exit of highly-rated midfielder Paul Pogba, who has gone on to make such a positive impact at Juventus.
Moyes does not believe Januzaj will be leaving as well, especially now he has started to make an impact at United following his match-winning double at Sunderland a fortnight ago.
"I'm confident," Moyes told reporters.
"I want to get it done and hopefully it will all get sorted.
"You're 18 and you are playing for Manchester United. I don't know where else would you want to play."
When Moyes was appointed as Sir Alex Ferguson's successor in the summer, the Scot knew he was joining a club with a rich tradition in the development of younger players.
Januzaj certainly caught the eye on United's pre-season tour of the Far East and Australia.
Moyes' initial intention was to integrate him into the United line-up earlier than he actually did.
"I would have played Adnan three or four weeks earlier but the circumstances didn't feel right to put him in the team for different reasons.
"He's a really good football player who is going to be able to play in several positions.
"Time will tell what his best one will be.
"He could arguably be a centre-forward - he's that good with his back to goal - but is also very good coming in off the left or right.
"As he showed at Sunderland, he has a goal in him as well."
Moyes regards Januzaj as "the most level-headed player I have come across".
Given he introduced Wayne Rooney and Ross Barkley to the Everton first team at a similarly tender age, it is some statement.
Yet Moyes has seen in Januzaj's willingness to turn out for United's reserve team this season, there is no massive ego to feed.
"He's got no ego to him," said Moyes.
"He has no problem training with the reserves or youth team.
"He's a real humble boy who has been well looked after by his dad and family.
"He can handle anything that comes.
"I've spoken to him about what he has to do.
"There are going to be dips at times with young boys and we'll try and look for that but I don't see it at the moment.
"Obviously I won't play him in all the games or use him all the time but we'll certainly consider using him when we think the time is right."
It is a selection policy that is bound to be greeted warmly by the Old Trafford faithful, brought up on stories of the Busby Babes and the famed 'Class of 92', three of whom are now part of the Old Trafford coaching set up.
"When I got the job, part of it was to introduce young players," said Moyes.
"When we're in the dressing room area I see Matt Busby's comment that if you're good enough, you're old enough.
"That has near enough been the psychology through this club.
"The last manager put young players in when he had them and if I've got more young ones, I'll look to play them."
It is the kind of wide-ranging attitude that made Moyes the stand-out candidate to replace Ferguson in the summer.
Yet he accepts that alone will not insulate him against poor results.
"I need to get results like any other manager," said Moyes.
"I'm not daft enough to know any different.
"I also know there's a long-term plan, which I'm part of.
"The board have told me about it.
"I'll take a few punches on the nose to get where I want to go.
"Hopefully, in the end, I'll be better for it."
Januzaj's father has said his son may spurn the advances of Belgium and Albania due to the unprofessional conduct of both nations.
The United youngster is eligible to play for several nations, with England also a possibility once he completes a five-year residency period.
And Roy Hodgson's hopes of persuading Januzaj to opt for the Three Lions may have been given a boost by Abedin Januzaj, who was speaking to Kosovo-based broadcaster KTV.
"I hear an official from the Albanian FA said they talked to me eight months ago, which is a blatant lie. They did a show called 'Patriot or Traitor' for my son and FSHF said they called him up. That is not true. They are not professionals,'' he said, in a transcript published by the Daily Telegraph.
"I am not closing the doors to Albania, because we are Albanians. However, they are closing the doors to themselves. I only deal with professionals on the issue.
"(My son) won't be engaged with Belgium because he is focused at United. They wanted him, the coach, for the game against Croatia. But, I don't like their approach either.''
On England, the senior Januzaj added: "I know Roy Hodgson said he would consider having Adnan in the English squad in the future. But, as I said for Albania and Belgium, it is not something we are not considering right now. Our full potential will be focused in doing well with Manchester United.''