Hammers have offered the Frenchman a two-year deal, worth more than £7.5 million.
They are prepared to pay the ex-Arsenal star around £75,000 a week to get him to their London Club.
Their joint-chairman David Sullivan said, "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."
As Avram Grant is set to be the Hammers' new manager, it will trigger a re-shuffle of team, with big names set to leave and some to join.
Sullivan also said, "We do not anticipate losing more than two first-team players."
"The idea is to sell maybe two and a few fringe players and bring in five new ones. We do not want to decimate the squad but rebuild it."
Thierry Henry has been playing at Barca since 2007, when he left another London club, Arsenal.
However, he may not be moving back to the Premier League as New York Red Bulls have also put in an offer for the star striker.
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