Kagawa, 23, joined the Red Devils from Germany Bundesliga champions Borussia Dortmund in Jun for £17 million. The Japan international has had successful spells in Germany after moving to the Bundesliga from J League's Cereza Osaka in 2010.
"I'm happy to be here and every day I'll take the chance to improve myself and move further up. It's definitely the greatest challenge of my career and I hope to learn and improve every day," Kagawa was quoted as saying in ESPN.
"I've always said I have to improve my playing qualities; now I've been here a bit I can really feel the difference. It's a top-level club and environment, and when the other internationals join us it will be even better," he said.
The 23-year-old, who can also play as a winger and second striker, will go on a pre-season tour of South Africa and China.
Kagawa said its an honour to get a chance to play at the top-level and he is eager to prove his ability with Manchester United.
"Everything here shows me it's a really big club and it's a great honour to play here. I want to really contribute by achieving concrete results."
Meanwhile, manager Sir Alex Ferguson hinted that Kagawa could make an immediate impact as Manchester United look to claim the Premier League title next season.
"He can give us that bit extra in the final third of the field. If his goalscoring record continues, he could be a very good player for us", Ferguson said.
Ferguson further added that the 23-year-old winger could start the season if his English improves over the coming months.
"I'm sure he'll improve his English over the next few months," said Ferguson.
"He can start the first league game quite easily. I've got no problem with that. We shouldn't be looking at one player changing the way we play but he could make a difference."