The Japanese international joined United last summer from German champions Borussia Dortmund for a fee in the region of £17m.
However, despite the fact that he has scored two goals during his 12 outings in all competitions so far, Kagawa has insisted that he is "not satisfied" with how he has performed.
"I'm not satisfied what I have done here at all," said the 23-year-old. "It is a big and really famous club in the world, but I need to overcome lots of pressures and be mentally stronger.
"Unfortunately I got injured after a couple of months, but I have to be better and grow as a player and as a person in terms of contributing to team's wins and succeeding here.
"We have a really strong squad with many class players, so it is lucky for me to have opportunity to train with them and play games with them, but I'm really feeling that I need to perform better."
Kagawa signed a four-year contract at Old Trafford when he arrived from Dortmund.