The 23-year-old former Borussia Dortmund star left Tokyo's Narita airport late Thursday and was due in Manchester for just 11 hours during a 44-hour, 22,000-kilometre (13,700-mile) whirlwind trip.
"It's just a medical check. I hope it will end with no problem," the attacking midfielder, who helped Dortmund to the Bundesliga title for two straight years, told Japanese media at Narita airport.
Kagawa, who moved to Dortmund in 2010 from J-League side Cerezo Osaka, said he would not have time to see his new club's facilities including their Old Trafford ground on his current trip.
Kagawa said his new boss Sir Alex Ferguson had promised to use him behind a central striker in a 4-4-1-1 formation, the major daily Asahi Shimbun reported.
United announced earlier this month they had reached an agreement with Dortmund on Kagawa's transfer which would be completed once he had passed a medical and obtained a UK work permit.
The deal is thought to be worth around 12 million pounds ($19 million).