The Ukrainian international’s contract is set to expire at the end of the season and the Eastern European giants are believed to be keen to sell in January rather than lose him for nothing during the summer.
A fee in the region of £1.6m is likely to be demanded, making the 27-year-old an ideal target for the Premier League club.
With Fabio Borini injured and Andy Carroll loaned to West Ham, the Reds have only Luis Suarez to call upon in a central attacking role.
The Uruguayan has been in inspired form this season, but there are worries that he cannot shoulder the responsibility of being the club’s singular source of goals as his supporting players continue to misfire.
They have been linked with the likes of Klaas-Jan Huntelaar and Edinson Cavani in recent times, but both would command huge investment in terms of wages and transfer fee.
Milevskiy on the other hand will be available at a bargain rate and be ready to help the club in the fight for Champions League qualification.
The forward has spent his entire career in the Ukrainian capital, notching up in excess of 150 appearances and finding the net regularly.
He has also won 50 caps at international level notching eight goals.