That is despite the fact the Italian club have accepted an identical offer from Serie A rivals Napoli.
Liverpool hope a deal can be done as Zielinski wants to join them, having indicated he is prepared to see out the remaining two years on his Udinese contract and leave on a free transfer.
Meanwhile, Mario Balotelli is back at Anfield after a loan spell at AC Milan.
The Italy striker, who signed for Liverpool from Milan in August 2014 in a £16m deal, returns to a much-changed squad following his season-long loan at the San Siro.
Liverpool manager Jurgen Klopp has made four major signings ahead of the new Premier League season, adding £34m forward Sadio Mane, goalkeeper Loris Karius, midfielder Marko Grujic and defender Joel Matip to his squad,
Balotelli returned to the San Siro on loan in August 2015 after a disappointing first season at Anfield, where he netted just four times in 28 appearances.
The 25-year-old started pre-season training with Liverpool on Saturday.
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Liverpool have had an £11.7m bid for Poland midfielder Piotr Zielinski rejected by Udinese.