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."