"Due to Dimitri's outstanding impact since his arrival at the Boleyn Ground, the club have initiated amicable and productive talks about an improvement and extension to his contract," West Ham said on their website (www.whufc.com).
"Those talks are progressing well and it is hoped they can be concluded shortly."
West Ham also made it clear that neither Payet nor his agent had made any demands to the club about a new contract.
The France international earns around 70,000 pounds ($101,000) a week with the Hammers, according to British media reports.
"We have to move and the most important move is to keep your best players and to add some new players who are needed," Bilic told British media.
"Dimitri Payet is our best player -- I have no problem whatsoever to say that. Of course, I would love to have him happy, long term, at the club."
West Ham host Liverpool in an FA Cup fourth-round replay on Tuesday before a league trip to third-bottom Norwich City on Saturday.
(Reporting by Shravanth Vijayakumar in Bengaluru; editing by Ed Osmond)
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