New United boss Jose Mourinho is desperate to land Mkhitaryan and has upped the ante in the battle for his services.
United had an offer of around £18m rejected by the German club following the end of the Premier League season.
But executive vice-chairman Ed Woodward has spoken to Mourinho about the attacking midfielder and decided to return with a second bid.
Sources close to the on-going negotiations reckon United are confident of making him their second major signing of the summer.
Mourinho has already landed Ivory Coast defender Eric Bailly from Villarreal in a £30m deal.
He is also expected to confirm the signing of Swedish superstar Zlatan Ibrahimovic on a free transfer once he returns from Euro 2016.
Now it seems Mourinho wants to add Armenian international Mkhitaryan to his squad as well before United embark on their summer tour of the Far East next month.
Dortmund have valued the 27-year-old at £30m, but United are confident of reaching a compromise with them after bidding close to the asking price.
Mkhitaryan, who scored 23 goals in the Bundesliga last season, has just 12 months of his current deal remaining at Dortmund.
He has refused to sign a new one after setting his heart on a move to Old Trafford, but Dortmund bosses remain determined to keep him.
Dortmund have already seen the likes of Ilkay Gundogan and Mats Hummels leave for new clubs this summer and can ill afford to lose Mkhitaryan as well.
But keeping an unsettled player is always a risk and now United’s sizeable bid has given them something to think about.
It is understood the bid includes add-ons that would take the deal past the £30m mark if he is a success at Old Trafford.
Mkhitaryan’s agent Mino Raiola said: “Manchester are Micki’s dream club, he wants to join them by any means. It would be a sin to stop talking.”
