The Hammers have signed Gianluca Scamacca this summer from Sassuolo but they want more talent from Serie A.
West Ham want to add to their midfield by signing Zielinski from Napoli, but their first offer for him was not sufficient enough for the Naples club.
As such, they are ready to make an improved offer for the midfielder and now they are making a push for his signature.
West Ham have hit the gas in their pursuit of Zielinski and are set to make fresh contact over a deal, according to Italian daily La Repubblica.
The Hammers will try to achieve a breakthrough shortly in the negotiations and close out an agreement with Napoli.
It was suggested that Zielinski is not entirely convinced of a move to West Ham, a similar problem being claimed with another Hammers target, Filip Kostic.
Zielinski missed just three games last season for Napoli in the league and scored six goals in addition to providing five assists.
