The Belgium-born 18-year-old marked his full debut for the club with a decisive double as United came from behind to beat Sunderland 2-1 at the Stadium of Light.
His performance prompted former England international Gary Lineker, who was watching the game at the BBC studios with England manager Roy Hodgson, to tweet that he could qualify for England in 2015.
Asked if the FA had been in touch with the club over Januzaj, Moyes replied: "Yes, they have."
Januzaj, who is of Kosovan descent, is eligible to play for several different countries already, Belgium, Albania and Turkey among them.
However, Moyes believes he could qualify for England in the near future because of residency rules.
He said with a smile: "And maybe Scotland as well. I think there is some way that he could qualify for England, yes.
"I think he's got a choice of countries to play for. He's got probably three or four different countries he could choose to play for.
"Obviously, he has been called up by Belgium several times and chosen not to at the moment, and he's wanted by, I think, Albania, Croatia and a few others as well.
"He can play for several countries - and England as well. I don't know how that is. I think it's something to do with his residency. That's something to do with it."
Just why Hodgson might be so keen to acquire Januzaj's services was apparent on Wearside as he dragged United back into a game in which they trailed after just five minutes and then fired them to victory inside a whirlwind six-minute
spell in the second half.
Craig Gardner had given the bottom-of-the-table Black Cats an unlikely lead after Phil Jones and Nemanja Vidic between them failed to deal adequately with Emanuele Giaccherini's driven cross, and it was one they deservedly held until minutes into the second half.
United got themselves back into the game when Januzaj expertly drilled home Patrice Evra's low cross, but his second finish was even better as he thumped a left-volley into the bottom corner after John O'Shea's 61st-minute headed clearance dropped to him.
Moyes said: "He's a very, very good player. But we will do everything we possibly can to keep his feet on the ground. We played him in the reserves at Bolton last Monday.
"That's what we will do at the club. He's a very humble boy, works hard, so we will do everything we possibly can to keep a level on it, but we know we have got a really exciting talent there.
"All young boys want to play for Manchester United. I don't know a better place for them to have the work, the coaching, a club that promotes young players - and I'd like to say a manager who promotes young players as well.
"If I was a dad of a young boy who was good enough, I would be saying, 'I think there's a chance of you getting in the Manchester United team because they push you forward and they promote their young players'.
"That's part of my remit and I was more than happy to play Adnan today."
Interim Sunderland head coach Kevin Ball was left to reflect upon a fourth successive home defeat for his team, who have now collected just one point from a possible 21 this season, but one which yielded more positives.
Asked if he felt the players were playing for him, he said: "The players are fighting for themselves - they need to fight for themselves as well.
"What I mean by that is they need to work hard for each other, and they know that. I will demand that from them and I would expect them to demand it from themselves.
"If people feel that's because of me, that's fantastic. But it's not about me, it's about the players in this club and it's important they do work very, very hard to give themselves the best chance of getting the results."