After helping the Hammers finish seventh in the Premier League with a club record 62 points on the back of his nine goals and 12 assists, Payet's star rose even further at the European Championship.
He scored three times as France finished runners-up to Portugal and he was also named in the team of the tournament.
Naturally the 29-year-old has attracted interest from several of Europe's big clubs like Real Madrid, but Gold has warned they won't entertain offers under £50m.
"The super clubs would come after a player of his ability," he told BBC Radio. "And of course we wouldn't even listen to any offer under £50m, and even then, as I said, we need to keep our best players.
"To be absolutely honest with you, we just don't welcome any offers at all for Dimitri.
"He is instrumental, he's important to us, the team is built around him. I would say to you he's not for sale."
While West Ham don't want to sell their star attraction, Gold admits that the lure of Champions League football could play a big role in Payet's decision to leave, but says the club is doing all it can to make him feel at home.
"You make him as happy as you can. He has a great relationship with the manager which is very, very important to a player," Gold said. "He's settled in London with his family, the fans adore him, and we're going to the Olympic Stadium.
"So we've done everything we possibly could do and I'm hoping that that will be enough."