Udinese slap a £15m price-tag on Liverpool's midfield target.
Zielinski is currently in France on international duty with Poland for Euro 2016.
The Empoli president urged the player to snub a switch to Anfield this week, according to an interview by Radio Kiss Kiss, but the Liverpool Echo report that he will not make a decision on his future until Poland's participation in the tournament has come to an end.
The report goes on to state that £9.5 million is nowhere near enough, with the Italian side demanding £15 million for his signature.
But the young midfielder is interested to work with Klopp, the Echo went on to suggest.
Liverpool are yet to do anything major this summer, signing goalkeeper Loris Karius after Joel Matip and Marko Grujic officially joined the club earlier in 2016.
They may be prove to be excellent additions, but supporters will be desperate for a little more box office about their future recruits this summer.