United are believed to be at the front of the queue to sign Kagawa, who is out of contract next summer and has refused to sign a new deal with the German champions.
The 23-year-old has been a driving force behind Dortmund's back-to-back title success and impressed again when scoring in last weekend's 5-2 victory over Bayern Munich in the DFB-Pokal final.
Ferguson attended the match and Kagawa has revealed he met with the United boss to discuss his future.
"It was good that I was able to meet him in person and talk," Kagawa said.
Kagawa has been widely linked with a £7 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."