The Sun reports that the Reds have been subject of a transfer ultimatum from the Ukrainian side.
Liverpool face competition from the likes of Bundesliga side Borussia Dortmund in the race for the 24-year-old.
Dortmund have reportedly made a £19.5million bid for Mkhitaryan.
Shakhtar Donetsk chief executive Sergei Palkin said: “I don’t know where Mkhitaryan is now, solely his agent knows this.
“Borussia offered £19.5m. I contacted Mkhitaryan’s agent on Tuesday. I wrote to him about the club’s position. We are prepared to sell for £25m.
“This offer stands for the next 10 days. If Mkhitaryan wishes to leave, they have to pay this amount for him.”
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The start of the 2017-18 Premier League season is just around the corner and is shaping up to be one of the most competitive yet.
As always it is the so called “big clubs” that have splashed the most cash in the transfer window so far this summer. Manchester United or City? Arsenal or Tottenham? Who are the winners and losers of the transfer window so far?