Negotiations have been on-going with the Japan international for a number of weeks.
The deal is subject to a medical and work permit, although this is thought to be a formality.
It is not anticipated the deal will be completed until the end of the month as Kagawa is currently on international duty and played in Japan's 3-0 World Cup qualifying round victory over Oman at the weekend.
"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."
Although the deal - which is expected to cost around £12million - has been anticipated for some time, it is good news for United.
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Yesterday [Monday], Eden Hazard explained why he thought Chelsea had "a better project" than the Red Devils, who lost their Premier League crown to Manchester City in agonising fashion last month.
Many United fans are privately concerned about the funds being made available to Sir Alex Ferguson and felt strengthening the midfield area was an absolute imperative.
Kagawa has already done enough in his two seasons in Germany to avoid the fate of Park Ji-sung, who was initially condemned as a "trophy" signing following his arrival from PSV Eindhoven in 2005, designed purely to sell shirts in the lucrative Far East market.
He was named in the Bundesliga's team of the season for 2010-11 despite being absent through injury for half the campaign.
Kagawa then went on to score 16 goals last term, when Dortmund completed the domestic league and cup double.
Ferguson and assistant Mike Phelan made a personal check on the 23-year-old the day before United's final match at Sunderland, when they saw Kagawa grab one goal and create another in Dortmund's 5-2 German Cup triumph over Bayern Munich.
Defender Rio Ferdinand was quick to offer his best wishes on Twitter.
"Welcome to Shinji Kagawa for his transfer to Man Utd," he said. "Here's to a succesful (sic) time here bro!"