Barcelona's Sport newspaper says an agreement has been struck between United and Dortmund over a €20 million fee. Hummels' move could be confirmed sometime this week.
Adding to the speculation is Matthais Ginter's imminent move from SC Freiburg to BVB as Hummels' replacement.
Hummels' impressive World Cup with champions Germany convinced United manager Louis van Gaal to push for his signing.
Barcelona were also keen, but were informed last week that the defender wanted to join Van Gaal at Old Trafford.
For the big stories and the banter follow us on Twitter: @tribalfootball