The two clubs are understood to be in advanced talks over a deal for the Japanese midfielder, with reports in Germany suggesting United have made an improved offer worth up to £17.6 million (€22m) including add-ons following an initial £13.6m bid lodged last week.
According to reports in Germany, Dortmund CEO Hans-Joachim Watzke has said that “both sides want to conclude the deal” while sporting director Michael Zorc told Kicker “the transfer is not done yet.”
Goal.com understands the main stumbling block is now Kagawa’s £100,000-a-week wage demands but United are confident of agreeing a deal by offering a lucrative commercial agreement, on top of a sizeable basic salary, that will see the 23-year-old earn money from image rights linked to Far East earnings.
Kagawa recently admitted to meeting with United manager Sir Alex Ferguson over a move and is certain to leave the German champions this summer as he has just one year remaining on his contract.
The Bundesliga outfit are resigned to losing the midfielder, and their CEO has confirmed that a move is close, telling Derwesten: "Both sides want to conclude the deal."
United had also been chasing Lille attacker Hazard but were forced to concede defeat in their pursuit of the Belgian when he announced via Twitter on Monday that he will be signing for Chelsea.
Goal.com exclusively revealed in April that United were lining up a bid to sign Kagawa and United will now step up their pursuit of the versatile attacking midfielder, who has scored 10 goals in 30 international appearances for Japan.
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.
Arsenal manager 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 29 goals in 71 appearances for Dortmund in the last two seasons, helping the club to two Bundesliga titles since joining from Cerezo Osaka.