The former Arsenal striker has been linked with a move away from Barcelona with reports indicating Major League Soccer side New York Red Bulls would be a possible destination.
However, West Ham joint chairman David Sullivan has revealed he wants to sign the 32-year-old and it is reported he is willing to offer him a two-year deal worth around £75,000 a week, more than £7.5million in total.
Sullivan told The Sun: "We're trying to assemble a more prolific strike force at West Ham and I'd love to have Henry here.
"Our club is in debt but a striker of his quality and standing would make it a great deal.
"Henry is still up there with the best.
"I've made an offer which would make him the highest-paid player in the club's history.
"And it's a serious offer because Henry could walk into almost any Premier League side."