The midfielder is currently on loan at Empoli and the 21-year-old is said to be likely to be a summer transfer target for manager Jurgen Klopp.
Klopp has already added Marko Grujic to his squad for the summer after a £5.1 million move from Red Star Belgrade, which saw the Serbian teenager loaned back to the club for the rest of the season.
The report from Goal suggests that Zielinski is one of the most sought after midfielders in Europe as they credit Manchester City and Manchester United with interest, while the Serie A trio of Napoli, Juventus and AS Roma have also been admirers.
The Polish midfielder has made 20 appearances this season and has found the net on two occasions, and was also on loan at Empoli last season where he featured on 30 occasions in all competitions - and has played 20 games for Udinese.
As for the transfer fee, the report says that a fee of 15 million Euros (£11.4 million) could land the player - who is credited for having strengths in distribution and dead ball situations, and is known to be pretty versatile. The report does also warn that the asking price could increase if he performs well at this summer's European Championships in France.
It would be an interesting move for Jurgen Klopp to go for another central midfielder, but dead ball situations have not produced much for Liverpool this season, and there have been times when they have been negligent in possession. If Zielinksi excels in these areas then the move would make sense but it would be a case of another young player at the club carrying the burden in an important position.
There are other players at the club who may be a little concerned for their futures at the club if this deal were to happen. Academy product Jordan Rossiter is out of contract at the end of the season, while Joe Allen's openly admitted that it has gone quiet when it comes to a new deal for him - although he has recently enjoyed an upturn in his form.