Payet is one of the best players in the Premier League and is in good form for France at the ongoing European Championship.
The attacking midfielder joined West Ham from Marseille in the summer of 2015, but the 29-year-old has already been linked with moves away from the London outfit.
Chelsea, Real Madrid and Paris Saint-Germain are all claimed to be interested in securing the services of Payet this summer.
According to a report in The Mirror earlier this week, Chelsea want to sign Payet, but the Blues have reportedly been told by their London rivals that the Frenchman will not be sold at any price.
Former Leicester and Derby midfielder Savage has suggested that West Ham co-chairmen David Gold and David Sullivan may have to sell Payet this summer.
"I know he signed a five-and-a-half-year deal, worth a reported £125,000-a-week , only four months ago, and I’m sure West Ham will move heaven and earth to make sure he leads them into their new home at the Olympic Stadium next season,” Savage wrote in The Mirror. “But there is no such thing as a bullet-prof contract in football any more.
“A long contract only offers clubs the protection of a big transfer fee if they cannot hang on to their prize assets. And if Real Madrid or Barcelona came calling for Payet, it would be a brave man who put his mortgage on him staying with the Hammers.
“Don’t get me wrong: I would love to see Payet sprinkling his magic on the Premier League next season and I hope he stays in England.”
Payet scored nine goals and created 116 chances in 30 Premier League appearances for West Ham last season.
The attacking midfielder has scored two goals and created 14 chances in two matches for France at the ongoing Euro 2016 finals.