Moore's first wife, Tina, who lives in the United States, has put the medal from England's 1966 World Cup victory and other memorabilia up for sale and there are fears the trophies could end up overseas.
Most of the money for the bid, expected to be more than pounds 2m, would come from the National Lottery. Robert Pratten, curator of the museum which plans to open next year, confirmed they had submitted an application to the Lottery's Heritage Fund to buy the collection.
An FA spokesman said yesterday: "As a museum their case is strong and we have offered our support. We also believe there may be a company or individual who would want to be associated with procuring the collection."
The FA has been examining various ways of ensuring that the trophies stayed in England and said it would be delighted if West Ham United, Moore's former club, succeeded in an application they have already lodged with the Lottery fund.
"The FA has one priority, and that is to keep the collection in England," the spokes-man said.Reuse content