"No doubt, it will truly be a big challenge for me, and I realise that mistakes will not be tolerated," the attacking midfielder told reporters in Japan, after returning from a World Cup qualifier in Australia.
"I have to work harder than ever. I have to take this step with that sort of resolve," the 23-year-old said.
The Premier League outfit announced on June 5 that it had reached a deal with the player and his German club Borussia Dortmund.
The agreement, estimated to be worth around 12 million pounds ($18.4 million), will see Kagawa become the first Japanese national to join Sir Alex Ferguson's men.
Kagawa revealed that Ferguson told him the transfer would be a huge challenge but wanted him to feel he could put his trust in his new manager.
"I was grateful to hear that from such a great person," Kagawa said.
The deal is nearly finalised, pending a medical exam later this month, Kagawa said.
Other big name clubs including Chelsea, AC Milan and Arsenal had been linked with Kagawa ahead of the signing.
In the end, he said: "Manchester (United) was the club that I most wanted to go to."
Kagawa cost Dortmund just 350,000 euros when he joined from J-League side Cerezo Osaka in 2010. He has scored 21 goals in 49 Bundesliga appearances for Dortmund, helping them win back-to-back league titles.
He has been a key player in Japan's national team, which drew 1-1 in the World Cup qualifier against Australia Tuesday night.
Japan lead the five-team Asian Group B on seven points from three games, followed by Iraq, Australia, Oman and Jordan.