The 27-year-old currently plies his trade in Russia with Dynamo Moscow, but is rumoured to be growing unsettled at the club.
Dzsudzsak is believed to have set his heart on playing in next season’s Champions League, and is worried that his ambition will not become a reality if his stays put.
Dynamo are currently close in terms of points to snatching a spot in next season’s edition of Europe’s top club competition, but with a difficult run in they look likely to miss out.
This has alerted Liverpool, who were linked with a £12m move for his signature last summer.
Brendan Rodgers has since remained keen on the 27-year-old, and may look to him as an alternative to Dnipro Dnipropetrovsk’s Yevhen Konoplyanka after a failed January move soured the relationship between the two sides.
Dzsudzsak has been lauded in the past for his lightening pace and sound technical ability, and is seen as an ideal candidate to support the likes of Raheem Sterling, Daniel Sturridge and Luis Suarez at Anfield.
Everton are also interested, but their local rivals’ likely ability to offer Champions League action is believed to be a deal breaker.