Kagawa, who struck 17 times for Dortmund last season, has only scored twice in his 15 appearances for United.
But the Japan international said that is because he is his own fiercest critic and expressed his disappointment that a knee injury has impeded his progress at Old Trafford.
He told United Review: "I haven't been satisfied with my performances so far but then I am always very critical of my own performances. Even worse I then got injured and had to go through the frustrations that come with getting injured in my first season with a new team."
Kagawa was sidelined for two months from the end of October but has returned to the United team and set up Wayne Rooney's first goal in Wednesday's 2-1 win over Southampton.
He added: "Now I'm just doing my best to catch up on what I have missed. There's a lot of pressure when you play for this club but all I can do at the moment is keep working hard every day to bring out the best in me."
Kagawa made his name operating just behind a lone forward, a role he occupied at the start of his United career. Recently, however, he has often been playing on the left of midfield.
He explained: "I have the most experience from my time at Dortmund in playing behind the striker. However, I just want to be part of the team, I will play wherever the manager wants to me to play. I am a versatile player and I want to play well in any position."