The Daily Star says Rodgers is keen to add Mkhitaryan, but has been told he must fork out £25m if they want the Shakhtar Donetsk ace.
The attacking midfielder has his heart set on Anfield and failed to report for Shakhtar's pre-season training camp in Austria.
The Ukrainian side's chief executive Sergei Palkin said: "I don't know where Mkhitaryan is. We are prepared to sell him for 30m (£25m).
"This offer stands for the next 10 days. If Mkhitaryan wishes to leave, they have to pay this amount for him."