The Japanese playmaker is understood to have agreed a four-year contract with the German giants, whom he played for before switching to Old Trafford in 2012.
A £5m fee, which may rise to £8m, has also been settled, with United prepared to take an almost 50% loss on the Asian superstar.
Kagawa is set to travel to the Bundesliga club to undergo a medical and put pen to paper over the weekend.
Louis van Gaal is understood to have made it clear to the silky passer that he has no future at United, as he sets about clearing out fringe players ahead of the transfer deadline.
Dortmund fans still value the 25-year-old highly, with some supporters having made petitions to get him back to the club in the not too distant past.
Kagawa signed for the Premier League heavyweights under Sir Alex Ferguson, and slipped away from the first-team during the reign of David Moyes.