"Manchester United is delighted to announce that it has agreed terms with both Borussia Dortmund and Shinji Kagawa for his transfer to the club," said a United statement.
"The deal is subject only to the player medical and obtaining a UK work permit. These conditions are anticipated to be completed by the end of June.
"Further details will be announced in due course."
The deal is expected to cost around £12million (18.4million dollars).