It has been reported by the British newspaper that the Hammers have made an offer of £14 million to sign the winger.
The London outfit’s initial bid of £12m was reportedly rejected by Marseille, and they have now increased their offer in order to recruit the France international this summer.
Payet is one of the best players in the current Marseille squad and was in good form for the French club in the recently concluded 2014-15 campaign.
The former Nantes star scored seven goals and created 134 chances in 36 Ligue 1 appearances.
The 28-year-old had a pass accuracy of 80% and a shot accuracy of 47%, and won three headers, 66 dribbles and 17 tackles.
The problem, though, is that Payet is 28 years of age and will have little sell-on value, which means that £14m for him looks a bit steep.