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."