Despite impressing so far this season, manager Louis van Gaal cannot guarantee the Belgium international first-team football and is keen for him to gain some experience elsewhere.
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Adnan Januzaj says he has signed for Borussia Dortmund
The 20-year-old was also a reported target for Tottenham, Everton and Southampton but appears to haver opted to move abroad.
Januzaj joined United from Anderlecht in 2011 and made his debut two years later against Wigan in the Community Shield.
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He has been in and out of the first team since, first under former manager David Moyes and now with Van Gaal at the helm.
He scored the winner against Aston Villa earlier this month but missed Sunday's defeat to Swansea, with United insisting he had a hip injury.