Following reports stating that the playmaker had flown in to the country, representative Thomas Kroth confirmed that his client held talks with the Red Devils boss
The 23-year-old is expected to leave the Bundesliga and DFB-Pokal winners this summer, with Dortmund sporting director Michael Zorc having revealed that his client has turned down a contract extension past his current deal, which expires in 2013.
And after reports emerged on Tuesday that the player had flown to Manchester, agent Thomas Kroth confirmed talks with Sir Alex.
Both Manchester United and Kagawa are interested in a transfer, Kroth told Kicker, but the representative revealed that "The outcome is open."
|5/2||Manchester United are 5/2 with William Hill to win the Premier League in 2013.|
Thus far, no other clubs have been closely linked with Kagawa, who is reported to be angling for an attractive offer from an English club.
Zorc, meanwhile, has confirmed that he is willing to listen to offers for the playmaker, with Dortmund reported to be holding out for a €15 million (£12m) bid.
Speculation linking Kagawa with a move to Old Trafford has been rife in recent weeks, with Sir Alex having observed the Japan international during Saturday's DFB-Pokal final victory over Bayer Munich in Berlin.
Kagawa was a key figure in Dortmund's title-winning season, scoring 17 goals and giving 13 assists in 42 matches in all competitions.