The Hammers have reportedly been tracking Zaza as a potential option as they search for a striker.
West Ham are said to be preparing a £15 million bid for the Juve forward, with Wolfsburg also interest.
But Zaza said: “I’ll assess my future after the Euros, but I hope to remain in Turin.
“I feel happy on a personal level, but not 100% because I think I can give more. I’m enjoying these Euros, I’m concentrated, fired-up and I hope to do well.”
West Ham are believed to have arranged talks with Juventus officials today, as they look to strike a deal.
The Hammers have made signing a striker their top priority this summer, with Zaza one of a number of targets who they have been linked with.
West Ham are also believed to be considering bidding for Michy Batshuayi, while they have had an offer rejected for Alexandre Lacazette.
Zaza is currently at Euro 2016 with Italy, where he has made two appearances.
The 24-year-old generally plays as a reserve option to Southampton forward Graziano Pelle and Eder for his national team.