The Reds have reportedly made the goalscoring midfielder their number one transfer target this summer but have suffered several setbacks in their £25m pursuit of the Armenian.
Spurs and Champions League finalists Borussia Dortmund have also supposedly registered late interest in Mkhitaryan, but The Mirror understands that he has settled on a move to Anfield.
However, his current boss Lucescu has warned the 24-year-old that he should not leave the Ukrainian outfit too hastily and perhaps wait for an offer from a bigger club.
"It is his decision, his desire, but there is still time for him to change his mind, because I think he is making a big mistake," said Lucescu.
"I can understand the urge to leave if he has a chance to go to a club that surpasses Shakhtar, like Barcelona or Real Madrid, but not many clubs are beyond our level - not clubs like Liverpool or Tottenham."
Mkhitaryan topped Donetsk's scoring charts last season with 25 goals.