The 22-year-old is currently playing for Poland at Euro 2016, having made his first appearance at the tournament in their final group game.
He could even be involved when they face Switzerland in the Round of 16 in Saint-Etienne on Saturday, hoping to set up a clash with one of Croatia or Portugal.
Reports from Italy had suggested the Reds had made headway towards sealing a £12 million deal with Udinese for Zielsinki, but there is no agreement currently in place.
Liverpool have had an initial £9.5 million bid turned down earlier this month as Udinese seek closer to £15 million for the Pole, though the Premier League side are confident an agreement can be reached.
Zielinski is keen to ensure he's fully focused on Poland's Euros campaign for the meantime, his agent Bartlomiej Bolek telling Goal: "He is completely focused on the Euros and nothing else now. Afterwards, there may be things to discuss."
But it is understood that Zielinski has already rejected the advances of Napoli. The Serie A side's president Aurelio De Laurentiis has already admitted that the player "prefers Liverpool" and "doesn't want to come" to Naples.
Liverpool made to wait on any Zielinski deal
Personal talks with Reds boss Jürgen Klopp are thought to have been a factor in helping persuade Zielinski to choose a move to England over remaining in Italy.
The versatile midfielder, capable of playing anywhere in the middle including in a No.10 role or even out wide, is understood to be intrigued by the opportunity of working under the German - who has a positive track record in developing young Polish players in his career.
Zielisnki spent the last two seasons on loan at Empoli, scoring five goals and providing six assists across 36 appearances in the Italian top-flight as they finished 10th last term.
His impressive season far from makes him a popular name to those unfamiliar with Serie A, but he is regarded as the right kind of player to add dynamism to Liverpool's midfield - which Klopp is looking to overhaul as part of his summer re-shaping.
Zielinski would become the Reds' second signing of the summer after goalkeeper Loris Karius, with Liverpool also welcoming January additions Joel Matip and Marko Grujic from Schalke and Red Star Belgrade into the squad.
The club remain keen on left-back Ben Chilwell of Leicester City, but are unwilling to meet his £10 million asking price, while they are desperately pushing to sign Borussia Monchengladbach midfielder Mahmoud Dahoud.
Sadio Mane is also one of Klopp's key targets as the club close in on a £30 million deal for the Southampton attacker, who the manager rates highly after scoring 15 goals in all competitions and adding nine assists in 2015-16.