Manager Sam Allardyce is desperate to bolster his ranks ahead of their return to the Premier League and has set his sights on strengthening the Hammers back four.
A season of consolidation is the mission for Allardyce to fulfil and he wants the 30-year-old Italian, who has won eight caps for the Azzurri, to spearhead their survival bid.
It’s thought they would face competition for Gamberini’s signature should Napoli firm up their interest although his representative Francesco Romano suggested it may not get to that stage.
“Alessandro has another year on his contract at Fiorentina and his priority remains the Viola shirt,” he told Julie Italia.
“We’ll talk with the club and evaluate what to do, because there have been plenty of approaches.
“Napoli? At the moment I haven’t heard anything from them.
“I have been contacted directly by two foreign sides – West Ham United and Bayer Leverkusen have asked about the player.
“But Alessandro is a Fiorentina player and I have said that they must speak with the club.”
West Ham have had quiet summer in the transfer market so far but did complete a deal to sign 25-year-old midfielder Mohamed Diame on a free transfer.
They had to beat off a host of European and domestic suitors to land the former Wigan man with Allardyce particularly excited about his new addition.
“His is potential is really good and he’s ready to go and compete in the first team straight away as he’s had three years in the Premier League already,’ Allardyce told the club’s official website.
“He’s 25 and can only get better and I hope he can go on to prove that to me and all of the West Ham United fans.
“He will be a very exciting addition to the squad.’