It is understood there has also been interest in the winger from Everton and Tottenham, Marseille and Valencia.
Januzaj has played for United this season, but boss Louis van Gaal appears happy for him to move to Germany.
The 20-year-old Belgium international joined United from Anderlecht in 2011 but did not make his first-team debut until 2013, against Wigan Athletic in the Community Shield.
Since then, he has been in and out of the first-team picture, under first David Moyes and then Van Gaal.
He scored the only goal in the Premier League victory at Aston Villa on 14 August, but he missed Sunday’s 2-1 defeat at Swansea, with United saying he had a hip injury.