As a dozen of the world's leading oil companies explore potential oil fields around the South Atlantic British colony, where the deposits may be one-and-a-half times the size of those in the North Sea, President Carlos Menem has issued a statement asserting that Britain should consider sharing sovereignty.
President Menem said that by 2000 he hoped "we will see the Argentine flag flying in the islands, either by itself or alongside other flags". He added: "If the Argentine flag flies alongside the British flag, that would be a step forward."
Although no firm geological data has confirmed the presence of economically viable quantities of oil, this has not stopped the island's 2,400 residents from making plans about how their resources will be marshalled.
During the licensing organisation period, the Falklands government estimated potential earnings from two hypothetical fields coming on stream. Yields of 250 million and 500 million barrels were forecast. This would net the islands an average of pounds 1.1bn a year over a 20-year period, the equivalent of pounds 483,653 for every man, woman and child in the islands.
Crucial to President's Menem's timetable is the fact that the revenue from oil, if any is found, would begin to flow in 2003.
Just as potential oil wealth eroded traditional ways of life in the Orkneys and the Shetland isles, so it appears to have affected the Falkland islanders.
Already the islands' economy has been overheated by the pounds 20m annual income from fishing licences granted around the island's ocean territory. Once thought to be a poor and remote community, they are now technically rich.
Control by Whitehall has eased since the Falklands War in 1982. The old dual masters, the powerful Falkland Islands Company and the Foreign Office, have been replaced by a semi-autonomous government.
The sheep, all 750,000 of them, are still there. But the population no longer describe themselves as sheep farmers. Fewer than 450 now live outside the capital, Stanley.
The British government and the Falkland islanders themselves will have been unimpressed by President Menem's comment. A spokeswoman for the Foreign Office rejected any idea of dual sovereignty and said: "We have no doubt of our sovereignty of the Falkland Islands. We are not prepared to discuss any change over the Falklands. There is no change to our position."
But while the British government says it does not intend to change its view of the Falklands, the people of the Falklands may be planning to change their view of Britain.
Although the islanders are still grateful for the intervention of the task force in 1982, which ended the attempt by the then Argentine leader, General Galtieri, to retake the "Malvinas", there may be a diplomatic feud between the islands and the mother country should oil be found.
The islands' council, the eight-person body that decides all policy, except defence and foreign affairs, has recently formally offered to pay the pounds 67m annual running costs of Mount Pleasant, the 2,000-strong garrison near the former 1982 military airport.
This sum remains Britain's sole financial contribution to the colony. When the Secretary of State for Defence, Michael Portillo, visits the Falklands early in the new year, as part of the celebrations to mark the 164th anniversary of Britain's formal adoption of the islands, he will bring with him a Treasury demand for the Falklands' figure to be agreed - crucially, before oil is struck.
Despite the old British convention that dependent territories are entitled to the wealth generated by their own resources, another Falklands' war, this time involving legions of international lawyers, could be in prospect.