The Premier League giants have been pressed into action after being made aware last week of growing interest in the attacking midfielder from Arsenal.
Reports emerged in Germany on Thursday that United had made a move for Kagawa and sources have confirmed to Goal.com that the club have made a formal bid for the Japan international.
United are believed to be hopeful of wrapping up a quick deal but it is understood that Arsenal's interest remains strong and the Londoners could also submit an offer for a key figure in Dortmund's back-to-back league titles.
Kagawa has only a year remaining on his contract and the Bundesliga club have hinted recently that they will not stand in the way of a transfer this summer.
Speaking last week, Dortmund sporting director Michael Zorc said: "I had a meeting with his agent and Shinji Kagawa has informed us that he will definitely not be extending his contract. Should we now receive offers for Kagawa, then we will consider and negotiate them."
Goal.com exclusively revealed in April that United were lining up a bid to sign Kagawa, whose reputation has grown over the last few months.
Manager Sir Alex Ferguson and his assistant Mike Phelan were then present at the DFB-Pokal final two weekends ago when Kagawa scored in Dortmund's emphatic 5-2 win over Bayern Munich.
Arsene Wenger is also an admirer of the Japanese midfielder, who arrived in Germany after the 2010 World Cup for a fee well below £1m and has been an influential presence in Jurgen Klopp's team.
He has scored 10 times in 30 international appearances, and his 71 appearances for Dortmund have yielded 29 goals, including the opener in the cup final, before setting up hat-trick hero Robert Lewandowski's first of the game.
Ferguson is aiming to free up space in the United squad by off-loading some of the dead wood. Michael Owen has been released and the club are also set to sell Dimitar Berbatov and Park Ji-Sung.