The Royal Naval College, which has been based at the site since 1873, moves out at the end of this year and the debate over what should take its place has been entangled in wrangles over leases and funding.
The decision in 1995 by Malcolm Rifkind, the former defence secretary, to sell off the 300-year-old masterpiece attracted such strong protest that his successor, Michael Portillo, was forced to back down.
Now the Greenwich Foundation for the Royal Naval College, responsible for finding an appropriate new use for the four main buildings, designed by Sir Christopher Wren, says there has, at last, been "significant progress".
The Foundation wants to sub-let three of the college's buildings, Queen Anne, Queen Mary and King William, to Greenwich University and a fourth, King Charles, to the Trinity School of Music. There are further plans for other historic parts of the site: a visitor centre in the Pepys Building (subject to National Lottery funding) and the creation of a Maritime Research Institute.
Sir Angus Stirling, chairman of the Greenwich Foundation, said: "I believe the area should continue to be what it has always been - a place of education and instruction in people's physical and moral well-being."
Last week, Sir Angus announced he was encouraged by talks with the Ministry of Defence (MoD), which is in effect the sole trustee of Greenwich Hospital - which owns the freehold of the site. The aim has been to negotiate a 150-year lease.
"Our exchange with the MoD forms the basis of a lease. It is a significant development and I'm now quite confident the terms of the lease will soon be sorted out." The MoD also confirmed that there had been a breakthrough.
Sir Angus said the Foundation was also "very close" to achieving an agreement with the Ministry for Culture, Media and Sport for funding repairs to the building.
The location of the Royal Naval College, on a broad sweep of the Thames, is among the most gracious in London. The buildings, now designated a World Heritage Site along with the nearby Royal Observatory, Cutty Sark and the National Maritime Museum, were built at the end of the 17th century as the naval equivalent of the refuge for Army veterans in west London - popularly known as the Chelsea pensioners.
The last naval pensioners left in 1869 and the Royal Naval College moved its offices from Portsmouth in 1873. Recent cuts have prompted the college to seek cheaper premises - two thirds of the college has already moved out. Those that remain are still accompanied by Jason, the 70-watt nuclear reactor used for teaching engineers for the nuclear submarine fleet.
Sir Angus has not been slow to note that, while the Foundation has sought pounds 10m to help it kick-start developments at Greenwich, the Millennium Dome, two miles down the Thames, has a budget of pounds 750m.
"The difference between funding for the Dome and the Treasury refusing to give us pounds 10m is incredible," he said. "It is peanuts and they couldn't even provide that much for what I believe are the finest buildings in Europe."
The decision to allow Greenwich University to move into the grounds has not met with unanimous approval. Admiral of the Fleet Lord Lewin last week questioned the University's ability to pay its way - a charge rejected by both the Foundation and Greenwich University.
The University is sensitive to accusations that it is somehow an inappropriate replacement for the naval college. John McWilliam, deputy Vice-Chancellor of Greenwich University, said: "We were founded as a polytechnic in 1890, when most current universities didn't even exist."Reuse content