Kagawa discusses United move with Ferguson
"It was good to meet and actually talk to him. He had some nice things to say to me," the 23-year-old told reporters.
Kagawa has been widely linked with a seven million pound transfer to the Premier League runners-up by British media, but said he was still considering his options.
"There's no timetable on my decision. I've been in touch with a number of clubs, and I can't say which one is the best for me yet.
"But I want to go to a team where it'll be challenging for me. It'd be nice if I end up playing in the Premier League."
Kagawa has enjoyed exceptional two years with the Bundesliga champions, who snapped up the talented youngster from Japanese second division side Cerezo Osaka in 2010 for only 350,000 euros.
Dortmund had hoped to tie the coveted player to a longer contract but the Japanese is set to reject that offer and move.
"The fans embraced me as if I'd been there for a decade and not just two years," he said of his spell in Germany. "I really felt loved. I was blessed to have the manager and team-mates I did. The coach told me I could come back any time I wanted to."
Kagawa had said previously that he would decide on his future after Saturday's German Cup final, where he scored the opener in Dortmund's 5-2 thumping of rivals Bayern Munich.