'It is a big challenge for me. I have nothing but a positive mind now,' the 23-year-old told reporters ahead of his departure on a flight from Osaka in western Japan.
"I want to fight with confidence," he added.
Kagawa, on a four-year deal from German champions Borussia Dortmund, is to attend a team practice and hold a news conference in Britain on Thursday.
Kagawa, having been ruled out of the Olympic football tournament in London, is set to feature in United's pre-season tour of South Africa and China.
"First, I want to display what I have at the training camp and prepare myself so that I can firmly gain a position in time for the opening game," Kagawa said.
The Japanese player's arrival is expected to be a boost for United, who were within minutes of retaining their Premier League title last season until a late winner on the final day for Manchester City in a 3-2 win over Queens Park Rangers saw their cross-town rivals crowned champions instead.
Kagawa helped Dortmund land successive German titles and scored 17 goals as they completed a domestic double last season.