In the first three days after the Heritage Department invited bids from prospective occupants, the company handling the sale of the 150-year lease, Knight, Frank and Rutley, was inundated with requests for brochures and further information.
The high level of interest comes amid growing concern over the preservation and character of the buildings, which many fear will be threatened once the Ministry of Defence hands over the keys.
Initially the estate agents have held discussions with middlemen wishing to clarify the restrictions surrounding the lease of the Grade I listed complex, dating from the 1600s.
However, while most of the companies or bodies which have shown an interest have not revealed their identities, the auctioneers are convinced they are serious players aware that stringent conditions will be attached to ownership.
One hurdle for any prospective buyer of the buildings, designed by Christopher Wren and described in the brochure as "an internationally known landmark adjoining the River Thames", is the annual maintenance bill of pounds 3m.
Equally problematic, though, may be finding an organisation large enough to fill the properties and meet the Heritage Department's criteria that the use should be "sympathetic to the character of the site" and allow the public access to the Painted Hall and the Chapel.
Michael Portillo, the Secretary of State for Defence, has already warned: "Tesco need not apply, nor need golf clubs or hotels or anyone like that."
A team from the National Maritime Museum has revealed that it is putting the finishing touches to a scheme which would see the area designated a World Heritage site.
The scheme would include the University of Greenwich renting academic and student accommodation, the Prince of Wales's Institute of Architecture leasing office space and several other shipping and maritime firms taking offices. Residences and staff flats would also be rented, and the museum would open the Painted Hall and chapel to the public, charging an entrance fee.
Sir Neil Cossons, an English Heritage commissioner and former director of the National Maritime Museum, recognises the importance of preserving the "special and precious" buildings.
"But the issue here is not about sale or lease, it's one of finding an appropriate use and providing some formula," he said. "Restrictive covenants in the lease must give greater protection than just its Grade I listing."