The Japan international scored six goals in 57 appearances for United.
"I had a real ambition to play in England - but now I feel I am back where I belong," the Daily Mail quotes Kagawa as saying. "Being back with Dortmund feels like being back with family - and as a player there is no better feeling than that.
"I don't feel like I have anything to prove to the fans in England - but I know that in England they did not see the best of me.
"Already I am feeling that I am on the way back to the level I want to be. It will take time - but I am finally on the right path."
Kagawa scored in Dortmund's 3-1 victory over Freiburg on Saturday.