Payet signed a five-and-a-half-year contract with the English Premier League club when he joined from Olympique de Marseille last year, but his performances this season have attracted interest from rival clubs, including some in the big-spending Chinese Super League.
Payet has scored six league goals and provided five assists in 18 games for the Hammers this season as the club has mounted an increasingly credible challenge for a top-six finish.
"I think we are very close to Dimitri signing a new contract," Bilic told reporters ahead of Saturday's league clash away to a struggling Norwich City side.
The manager also spoke out in support of cheaper tickets for fans after a mass walkout at Anfield at the weekend resulted in Liverpool's owners backing down on plans to implement a significant increase in prices next season.
"Football is not polo or golf. It is not a sport for the upper class," Bilic said. "It's about a balance. In the Premier League you get a lot of money from TV and sponsorship, so tickets should be a good price for fans."