The 23-year-old is United's first high-profile signature of the close-season with his switch from German champions Borussia Dortmund and the move has generated huge media attention, especially in his native Japan.
"I'm a member of a great club at Manchester United," Kagawa told the club's website.
"It's definitely the greatest challenge of my career and I hope to learn and improve every day.
"I think I can handle the pressure and am pretty confident I can adapt to the style of the Premier League.
"I definitely feel the attention of the media globally now."
Kagawa made his name at Dortmund as an attacking playmaker where he made 49 appearances and scored 21 goals in two glittering seasons that yielded back-to-back Bundesliga titles.
His new manager Sir Alex Ferguson said he expects Kagawa to adapt quickly to life at the club and predicted his impact would be felt early on.
"Once he's adapted and gets to know some football phrases our players will use, he'll be used to that. He can start the first league game quite easily, I've got no problem with that," Feguson said.
"He can give us that extra in the final third of the field. If his goal-scoring record continues he could be a very good player for us."
Ferguson did not discount more new faces arriving before last season's Premier League runners-up begin the new campaign season against Everton on August 20.
"We could possibly bring one or two more in," he said."
"When there's a major international tournament on, there's a delay in the transfer market. But now the European Championship is over, we're trying to bring one more player in."
United begin their pre-season tour on July 15 where they are scheduled to play matches in South Africa, China, Norway and Sweden.
