The Japan international, 23, is reportedly set to cost United £14 million and will join the Premier League club after passing a medical and obtaining a work permit.
A club statement read: "Manchester United is delighted to announce that it has agreed terms with both Borussia Dortmund and Shinji Kagawa for his transfer to the club.
"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."
United missed out on Belgium international Eden Hazard, who completed his £32 million transfer to Chelsea on Monday, but the imminent signing of Kagawa gives them some much-needed creativity in midfield.
His two-year spell in the Bundesliga saw Kagawa become one of German football's most impressive playmakers.
Kagawa signed for Dortmund from Cerezo Osaka for a fee of just €350,000 (£280,000) in the summer of 2010 and helped the German club secure the Bundesliga title in his first season at Signal Iduna Park.
The midfielder again played a central role in the club's double success this season, scoring 17 goals and claiming 10 assists in 42 games in all competitions.
Kagawa's refusal to sign a new contract with Dortmund saw him become the subject of constant speculation towards the end of the 2011-12 season and he made clear to Dortmund that he wanted to pursue a move to England.
When Kagawa completes his move to Old Trafford he will become the fifth Japanese player to play in the Premier League following Junichi Inamoto, Kazuyuki Toda, Hidetoshi Nakata and Ryo Miyaichi.