The striker is ready to leave Turin this summer after falling down the pecking order and making just 19 league appearances last season.
The Italian had been set to join Wolfsburg on loan, but the transfer has collapsed during the final negotiations.
Now, according to reports in Italy, Juventus have offered West Ham the chance to land Zaza this summer.
The Hammers are in desperate need of a striker before the transfer window shuts after being hit by injuries and they are keeping tabs on Manchester City forward Wilfried Bony.