The 27-year-old midfielder has emerged as one of Jose Mourinho’s key transfer targets in recent weeks and the deal could now be finalised early next month.
Hans-Joachim Watzke, Dortmund’s chief executive, has said that the Bundelsiga club are keen to settle the matter before the start of their pre-season training on 4 July.
"We will assess his situation this week," Hans-Joachim Watzke, the club’s chief executive, told kicker.
"We not only need to discuss it with [the club's sporting director] Michael Zorc and [manager] Thomas Tuchel, but also with the bodies," Watzke said. "It is a difficult decision."
According to Raphael Honigstein, the German football writer, Dortmund have sent an official to negotiations in Manchester and the deal is close to completion.
Despite the fact that Mkhitaryan has just one year remaining on his deal at Signal Iduna Park, it is expected that he will cost United around £27million.
The Armenia international could replace Juan Mata, who may leave Old Trafford this summer following Mourinho’s appointment.
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Mourinho sanctioned the sale of Mata to United while manager of Chelsea in 2014 and it is thought that Mkhitaryan would take the Spaniard’s place on the right-hand side of midfield.
Mkhitaryan scored 21 goals in 50 appearances for Dortmund last season, his best return since joining the club from Shakthar Donetsk for £21.5m in 2013.
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