Southampton are also thought to be interested in the Juventus forward.
Juventus' Simone Zaza celebrates after scoring.
Zaza is expected to leave Juventus this summer despite only joining the Italian champions last summer.
It has been a frustrating year for him though, as he finds himself down the pecking order behind the likes of Paulo Dybala and Alvaro Morata.
He managed just 24 appearances in all competitions, with sixteen of those coming off the bench.
Southampton manager Ronald Koeman
He totalled just eight goals in those appearances, but has clearly done enough to impress Southampton and West Ham, who the Mirror claims are interested in his services.
However, it seems he would be an expensive signing for either side, with Corriere della Sera reporting that Juve have set an asking price of as much €25million, around £19million.
Juventus' Simone Zaza celebrates with teammate Leonardo Bonucci
That would represent decent value for Juve, who spent £13 million to sign him permanently from Sassuolo last summer.
Whether either side is willing to pay it remains to be seen, although it is well know that the Hammers are desperate to add a striker to their squad this summer.
West Ham manager Slaven Bilic