The Arsenal midfielder spent last season on loan at Bournemouth in a bid to secure regular game time.
He made 29 Premier League appearances for the Cherries last term, but his injury curse continued as he suffered a broken leg in April.
Wilshere, 22, has just 12 months remaining on his contract at the Emirates but doubt has been cast over his future at the club.
The Hammers have been linked with a move for the England international this summer in a bid to help resurrect his career.
But West Ham chief Gold has played down their interest in the midfielder by outlining that the club’s priorities lie with signing two strikers.
When asked about the link to Wilshere, Gold told talkSPORT: “We are interested in all good players but we have to face the facts here – we have got good midfielders.
“We have got a good defence, we have got good midfielders, we have got good wingers.
“We need strikers and that is the key to everything for this season so we are focused first and foremost on bringing in two strikers.”
Wilshere’s Arsenal team-mates Olivier Giroud and Theo Walcott have been mooted as possible targets alongside Manchester City’s Kelechi Iheanacho.
And Gold insists they are exactly the type of players Slaven Bilic wants to bring in.
“It’s clear to most fans that we are desperate for centre-forwards,” Gold continued.
“Andy Carroll is injury-prone as we know, we have had an injury problem with [Diafra] Sakho and we have just got to bring in centre-forwards. We have got to find the money.
“Without strikers, you struggle in the Premier League. You struggle in any division because strikers are your key players.
“You have to spend your money wisely but it must be on strikers, particularly strikers that have had experience with Premier League football – the very best we can afford.”
