According to La Gazzetta, the Old Lady are said to be unwilling to sell as they see Sturaro as a key member of their squad and big part of the club's future.
Is Sturaro snub such a bad thing?
But is the snub such a bad thing for either club?
After all Slaven Bilic's West Ham are just four points off the Champions League places and have a very decent squad with a host of top injured players returning to full fitness for the business end of the season.
Bilic is well stocked in midfield too with the likes of Barcelona loan star Alex Song struggling to get a game.
It could be argued that Ronald Koeman's Saints could do with some fresh impetus in the middle of the park as the club struggles to live up to the high expectations they build up following their magnificent campaign last season.
Southampton's wealth of midfield talent
The Dutchman's side currently sit 13th in the Premier League table and look on course for a midtable finish.
Both clubs should have more than enough talent to see them through the season comfortably even if Sturaro is a class act.
Could a bid be resurrected by the Hammers in the summer, though, with Bilic linked with a keen interest in Sturaro's Juventus and Italy teammate Simone Zaza, as reported by the likes of the Evening Standard.
Sturaro's former teammate Angelo Ogbonna is also with the Hammers too.