The Hungarian winger has been on the radar of the Reds for some time, and was linked with a summer switch to Anfield.
Rumours of a fresh January swoop have been circulating, but now the Anfield side face competition in the shape of Fulham.
New Cottagers chief Rene Meulensteen – who succeeded Martin Jol over the weekend – has already hinted that he may dip into the transfer market this winter, and is believed to be ready to bolster his side’s attacking line.
The London club are currently in danger of being embroiled in a season-long relegation scrap after picking up just 10 points from their opening 13 league games.
Reports suggest that Meulensteen is ready to lodge a bid for Dzsudzsak, who may be allowed to leave the Russian capital following the arrivals of a number of players during the last window.
Liverpool are also in the hunt, with Brendan Rodgers thought to be targeting a new wide-man to help his side maintain their charge for a top four finish.
But, Fulham will be able to offer the former PSV Eindhoven star a greater chances of first-team action, which may be a deciding factor.