Sky Sports reports that Januzaj is close to joining Dortmund on loan from United.
Reports have also linked the winger with the likes of Everton, Tottenham, Marseille and Valencia.
United boss Louis van Gaal is reported to be happy for the 20-year-old Belgium international to move to Germany.
Januzaj has failed to become a regular at Old Trafford since Van Gaal replaced David Moyes as manager.