The idea is being proposed by Robert McDowell, the founder of the arts centre Summerhall, ahead of the 75th anniversary of the International Festival, the Fringe and the film festival this year.
There is no official archive in Edinburgh encompassing the city’s world-famous cultural celebration, which is the biggest annual event on the planet.
Mr McDowell said opportunities to create a new Festival Museum in Edinburgh had been missed in recent years, at the site of the former Sick Kids Hospital, near Summerhall, and in St Andrew Square, close to where a new venue will be created for International Festival concerts.
Glasgow University holds one of the biggest collections of material relating to the Fringe as part of its Scottish theatre archive, while film footage is held at the National Library of Scotland's Moving Image Archive in Glasgow.Summerhall was created in Edinburgh University's former vet school after it was acquired by McDowell in 2011.
The year round arts centre also stages one of the biggest programmes of shows and events of any venue at the Fringe. In recent years it had been home to the official archive of the artist, collector and promoter Richard Demarco, who has been involved in Edinburgh’s festivals ever year since their inception.
Mr McDowell said: “This year is the 75th anniversary of the Edinburgh Festival. It is long past the time when the city should have put its back into the idea of having a museum dedicated to the history of the Festival.
“It has made a contribution to the world on a par with what we think of as the Scottish Enlightenment through its internationalism and the number of tickets that are sold every year.
"If any other city wanted to have the Edinburgh Festival they would have to put up more than £100 million. A lof of the annual investment in Edinburgh comes from foreign governments paying for companies and artists to perform here.
“The council and the Scottish Government get away with putting in around £10 million.
"We have got more than enough stuff here at Summerhall to kick-start a museum given what is in the Demarco Archive.
"As soon as a museum was started, however small, I just know that there would be contributions from thousands of people.”
He later failed in a bid to buy the historic home of Edinburgh's Sick Kids Hospital from NHS Lothian after setting out plans to turn it into a new gallery dedicated to children’s art.
Mr McDowell says the site would also have been able to accommodate an Edinburgh Festival Museum, but the opportunity was lost when it was sold to a housing developer.
He added: "The museum could have been created at the former Sick Kids Hospital, which was a sacred place more important to the people of Edinburgh than the castle. It was not a commodity.
“There were several buildings on St Andrew Square which might have been good for a Festival Museum, but they are gone now. However there will be other potential buildings around.
"Recent developments which the council has been involved with have not been concerned with culture at all.”