The Borussia Dortmund midfielder is a player Mourinho has earmarked, with United already having failed with an initial bid earlier this month.
Now Mourinho has sanctioned an improved offer, one which he is confident will succeed given that the 27-year-old Armenian international is out of contract with the Bundesliga club in 12 months.
The Germans reportedly want £30 million for their asset but the player - and his agent - have been outspoken in his desire to move to Old Trafford and Mourinho is confident he will soon join Ivory Coast defender Eric Bailly, signed in a £30 million deal from Villarreal, in a new-look United squad.
Mourinho will want Mkhitaryan, who scored 23 goals in the Bundesliga last season, well ensconced in his squad by the time United visit China for a pre-season tour in mid-July.
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And Swedish legend Zlatan Ibrahimovic already appears on course to announce a one-year deal with the Reds once his country’s interest in the Euros is extinguished.