The Hammers have been busy of late, appointing new manager Slaven Bilic last week before snapping up Sampdoria midfielder Pedro Obiang just a day later.
Croatian boss Bilic will now be hoping to improve his squad even further, and the Daily Mail believe that Marseille star Payet is a top target.
The 28-year-old has earned 14 caps for France during a successful career in Ligue 1, spanning spells with Nantes, Saint-Etienne, Lille and now Marseille.
With 21 assists and seven goals last season, Payet was a key player under Marcelo Bielsa, but is considering leaving Marseille with the club slashing wages after missing out on a Champions League place, according to the report.
That leaves West Ham hoping to lure Payet to England, with a new and improved £12m bid on the table - as well as a drafted contract worth £70,000-a-week over four years.
Such an offer is allegedly tempting for Payet, who could leave despite his strong relationship with manager Bielsa - and whilst Marseille could open talks over a new contract, West Ham are bidding to prise Payet away this summer as they look to improve a front line which struggled in the second half of the Premier League season.