They report that their newly appointed manager Vincenzo Montella is a fan of the Poland international but that Napoli are also interested in the midfielder.
Liverpool have reportedly tabled a £9.5 million bid for Udinese midfielder - who spent the last two seasons on loan at Empoli - but they are currently holding out for a fee closer to £15 million, according to the Liverpool Echo, although a compromise could be reached between the two parties.
The same Echo report suggested the Reds will have to wait until the conclusion of Poland's Euro 2016 campaign to wrap up a deal for the 22-year-old. They are in quarter-final action tonight against Portugal.
It seems that if Liverpool want their man then they are going to have to get a deal done relatively quickly in order to fight off competition and for the midfielder to play as full a part as possible in Liverpool's pre-season training schedule. Obviously he will be reporting to training later than others given Poland's run in the tournament.
It seemed for much of the summer that Zielinski has been keen on a move to Anfield more than anything else, having previously spoken glowingly of Reds boss Jurgen Klopp, and wanting to play with Liverpool play-maker Philippe Coutinho, as the Echo reported back in March.