The Japan international, who arrived at Old Trafford from Borussia Dortmund for a reported £17million in July, has struggled to produce the form that turned heads in the Bundesliga.
Injury has frustrated the 23-year-old's progress, with a knee injury suffered in October ruling him out for almost 10 weeks.
Kagawa, who was substituted early in the second half of United's 1-1 draw at Tottenham on Sunday, knows he has to improve.
"I'm really feeling that I need to perform better," he told the Daily Mirror. "I'm not satisfied with what I have done here at all.
"It's a big and really famous club in the world and I need to overcome lots of pressures and be mentally stronger.
"Unfortunately I got injured, but I have to be better and grow as a player and a person, in terms of contributing to the team's wins and succeeding here."