France Football claim that West Brom are the front-runners to sign the 29-year-old, who netted 23 goals in 39 games last term, although the East London club have long been interested in him, as per the Mirror.
The Baggies have already offered him a sizeable contract, and Gignac himself is keen on moving to the Premier League, although he was holding off choosing a future club until the Hammers appointed their new manager.
Marcelo Bielsa was linked with the vacant managerial seat, and Gignac was reportedly keen on linking up with his Marseille coach at the Boleyn Ground if he was appointed by the Irons.
However, Slaven Bilic was named as manager, but West Ham retain their interest in him despite the managerial appointment, with the Mirror claiming that he is looking to wrap up a deal for the forward in the coming days.
This is a positive update for the Hammers, who looked as if they might miss out on a real top class forward this summer, with the club still keen on moving for the type of player who can help usher in a new era of success at the Boleyn Ground.