Payet enjoyed a successful season in the Premier League in 2015-16 and is playing well for France at the ongoing 2016 European Championship.
According to The Sun, Chelsea and Inter Milan are interested in signing the 29-year-old attacking midfielder in the summer transfer window.
The London Evening Standard reported last month that West Ham fear that Payet could force a move to Spanish giants Real Madrid this summer.
West Ham co-chairman Gold has made it clear that the club will try their best to hold onto the Frenchman, but he has admitted that a move from Madrid or Barcelona could be too tempting for the attacking midfielder this summer.
“We’ll fight tooth and nail to make sure Dimitri pulls on a claret and blue shirt next season,” Gold told The Sun.
“We want to improve our squad and Dimitri is an integral part of what we’re trying to do. But we also must be realistic.
“If Barcelona or Real Madrid made a bid and he wanted to move there, it would be difficult to keep him.”
Payet joined West Ham from Marseille in the summer of 2015 and was an instant hit in the Premier League.
The 29-year-old scored nine goals and created 116 chances in 30 Premier League matches for the Hammers last season.
In four matches for France at the ongoing Euro 2016 finals, Payet has scored two goals and created 19 chances.