The 26-year-old, who currently plays for Russian outfit Dynamo Moscow, was believed to be a 'last-minute' alternative to Thomas Ince, whom the Reds were said to be chasing during the recent transfer window.
Inter Milan have also been named as potential suitors.
Dzsudzsak's agent Josef Vorosbaranyi said that his client is happy at the Arena Khimki, but revealed that he desires a move to one of Europe's "major" leagues, and has not given up hope of securing a move.
"Inter [Milan] along with Liverpool wanted to add Balazs to their squad, but Dynamo would not let the player go," The Metro quotes Vorosbaranyi as saying.
"My client really likes Moscow, but he wants to move to one of the major European leagues.
"I hope in the summer, Dynamo would look at doing a deal if an offer arrived."
Dzsudzsak has won 50 caps for his country.