According to the Daily Mirror, Dortmund have knocked back an opening £19m offer from Manchester United for the Armenia international, who is about to enter the final 12 months of his contract.
Mkhitaryan has rejected a new four-year deal on offer at Dortmund and German paper Bild claim Mourinho is willing to double his pay to £7.9m (€10m) per season.
The 27-year-old favours a move to United, according to his agent Mino Raiola.
“Man United is a unique opportunity,” Raiola told Bild.
“Micki wants to go there immediately.”
According to reports, Arsenal have also submitted a bid.
However, Dortmund are planning to keep the attacking midfielder for the final year of his contract before allowing him to leave on a free transfer.