According to the Corriere dello Sport, the Hammers are fascinated by the prospect of nabbing the Croatian centre-forward.
Kalinic was a flop the last time he played in England, being acquired by Blackburn Rovers in 2007, only to net just seven league goals before leaving.
The 28-year-old is also wanted by Tianjin Quanjian, whose recent interest has ramped the Croatian's price to around the €50 million mark, his release clause.
The Hammers will be getting rid of striker Simone Zaza in the January window, with Valencia his most likely destination.
They may baulk both at the price, and the fact that signing players from mainland European hasn't been a resounding success for them so far.
Kalinic has netted 20 Serie A goals since joining the Viola at the beginning of last season. He was one of that campaign’s early sensations, scoring ten times before the Christmas break.
Things took a turn for the worse in the second half of the season. Despite having managed eight strikes this season, only two of them have come at home.
This could be a problem for West Ham, who have struggled at the London Stadium since moving there this summer.