General Register House at the east end of Princes Street, designed by Robert Adam and dating back to 1788, is one of the oldest purpose-built archives in the world.
But an estates review by the National Records of Scotland (NRS) has concluded the landmark building, together with its neighbour New Register House, should not be part of its long-term future.
Instead, the NRS has said it wants to move staff and records to its storage facility in Sighthill.
The move comes as major redevelopments are planned in the immediate area, including the creation of the new St James Quarter and a revamp of the former Royal Bank of Scotland headquarters in St Andrew Square.
Conservationists warned any plans for the future of General Register House must respect its historical and architectural importance.
The building, with the statue of the Duke of Wellington immediately in front, features a stunning rotunda 50ft in diameter and 80ft high, inspired by the Pantheon in Rome.
It also houses the Scotland’s People family history centre, which opened in 2008. One user of the building said: “I can’t understand how they can even contemplate getting rid of a building of such national importance.
“It’s like saying ‘Edinburgh Castle is in a prime site – let’s hand it over to the nearest hotel chain’.
“General Register House is not just an Edinburgh landmark, it’s of national importance and it was custom-built for keeping records.
“There are plans afoot for a lot of redevelopment around that area. What would they do with General Register House? Would it just become another pub?”
Lothian Labour MSP Sarah Boyack said: “It’s important that landmark buildings such as General Register House are not just properly protected but that we see the best use of such buildings for the long term.”
Marion Williams, director of the Cockburn Association, said she hoped a positive outcome could be secured.
“We will want to work with all concerned to achieve the right outcome for these buildings,” she said.
“Let’s hope listed status has some meaning in this case. We must move with the times but be faithful to our heritage. It can be done.”
A National Records of Scotland spokeswoman said: “There are no immediate plans for National Records of Scotland to move out of General Register House or New Register House.
“Our long-term aspiration is to locate the majority of our staff in a fit-for-purpose facility in Edinburgh, and to expand and improve our archive and public facilities at Thomas Thomson House in the west of the city. This intention remains subject to a number of challenges and constraints, not least funding.”
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