The midfielder said the Scot, one of football's most successful managers, had "good words" to say about him during two hours of talks in Manchester this week.
Kagawa, who still has a year to run on his contract with the German champions and has stalled over signing an extension with the club, said he wanted to "think hard" before deciding where his future lay.
"It was good that I was able to meet him in person and talk," Kagawa told reporters on his return ahead of Japan's friendly against Azerbaijan.
Ferguson "had some good words," the player added when asked by journalists at Tokyo's Haneda Airport about the meeting, without disclosing what they discussed.
The 23-year-old scored 17 goals this season in all competitions to help Dortmund secure a second consecutive Bundesliga title and win the domestic double for the first time in their 103-year history.
He has made no secret of his desire to move to England with United tipped as front-runners to sign him.
Ferguson watched Kagawa score in Dortmund's 5-2 German Cup final win over Bayern Munich at Berlin's Olympic Stadium Saturday.
"I want to think hard before making my decision," Kagawa said, adding that several other Premier League clubs have also expressed interest in him.
"I want to choose a team where I can take on challenges," he added.
Dortmund are in a position to make a handsome profit by selling Kagawa, who cost the German club just 350,000 euros ($450,000) from Cerezo Osaka in 2010 but is now worth around 13 million euros.
Kagawa thanked Dortmund for allowing him to grow as a player, and said he will miss the thunderous roar of home fans and said coach Jurgen Klopp has told him he will always be welcome at the club.
Other big name clubs, including Chelsea, AC Milan and Arsenal have also been linked to Kagawa, media reports have said.
Source: AFP
Source: AFP