United said in a statement they expected the transfer to be completed by the end of June and have already agreed terms with the attacking midfielder, who has been a revelation in Germany over the past two seasons.
No details of the fee were released but German media reported that Kagawa, who has a contract with Dortmund until 2013, would cost United 17 million Euros.
"We would like to thank Shinji for his effort and his large contribution in two extraordinarily successful years," Borussia sports director Michael Zorc, who was responsible for bringing the player to Dortmund in 2010, said in a statement.
The 23-year-old attracted few headlines when he joined Dortmund for 350,000 euros from then Japanese second division club Cerezo Osaka but he surprised many by notching 21 goals in 49 league games in Germany.
He won two consecutive Bundesliga titles as well as the German Cup last season, helping Borussia establish themselves as the new force in German football.
His creativity and darting runs from the middle of the field will bring an added dimension to a United side that brought 37-year-old Paul Scholes out of retirement last season to help their Premier League title challenge.
With several midfielders injured, Scholes, who has agreed to play on next season, ended his six month retirement and returned in January to play a key role in helping them finish second in the Premier League behind Manchester City.
Kagawa showed his versatility for Dortmund this season by replacing injured Mario Gotze after the winter break in a more advanced playmaking role where he proved to be equally effective. He also boasts a strong goal-scoring record, netting 13 times in the league last season and setting up another nine for his team-mates.
United had been strongly linked with Kagawa for months with manager Sir Alex Ferguson meeting the player in Berlin immediately after Dortmund's German Cup victory over Bayern Munich in May.
The transfer, if completed, will be a welcome boost for the 19-times English champions after missing out on Belgium playmaker Edin Hazard who signed for rivals Chelsea on Monday.
Kagawa is currently with the Japan squad who are in the middle of the fourth round of their World Cup qualifying campaign. Japan face Jordan at home on Friday, then Australia in Brisbane on Tuesday after easily beating Oman 3-0 on Sunday.
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