The 22-year-old was attracting interest from Liverpool and AC Milan, but Napoli have won the race to sign the player.
Zielinski spent the last two seasons on loan at fellow Serie A side Empoli and has made a total of 82 appearances in Italy's top flight.
He becomes the second Polish player to sign for the Naples club this week, after Arkadiusz Milik signed for Napoli on Tuesday.