Kagawa, 24, struggled with injuries in his debut season at Old Trafford, scoring six goals in 26 appearances across all competitions.
The Japan international was outstanding for Dortmund in the 2011-12 campaign, helping them to a league and cup double, but largely failed to replicate his form following his move for a reported transfer fee of £13million.
And Kagawa has admitted he would like to play for Dortmund again, once he has proven himself with United.
Kagawa told TV Tokyo: "Yes, one day I will (return to Dortmund). If Dortmund are ok with it.
"I'm happy we won the league (last year) but I'm not satisfied with my performance.
"The club has a long history and it was great that I could play in Sir Alex Ferguson's last season.
"At first there were also moments when I felt a bit lost but I tried not to give too much importance on such a feeling."
Kagawa has described the 2013-14 season as "really important" for himself, as he bids to become a regular starter under new United boss David Moyes.
But competition for places is high, with Kagawa – who was often employed on the wing last season – fighting for a spot alongside Ashley Young, Antonio Valencia and new signing Wilfried Zaha.
"I have no excuse any more after a year in the Premier League," he said.
"I know I'm at a place where you can't play if you don't achieve results. I'm determined to do it, also because there will be World Cup.
"It's really an important year for me."