The huge but harmless plankton-eaters, which grow to more than 30ft and weigh five tons, are at risk from over-fishing. Their fins are the ones most prized by the international trade for shark-fin soup.
More than nature conservation is now at stake. The basking shark is clouding the sometimes troubled relations between the semi-independent island and Britain.
Surveys have found a rapid decline in the number of basking sharks visiting Isle of Man waters. But while the Manx government forbids fishing for the sharks within its 12-mile limit, the UK government is refusing to support wider moves to control and monitor the global trade in shark products.
The Manx government has formally requested the Department of Environment in London to propose to other nations that the shark be covered by Cites - a treaty which regulates the trade in endangered species. The department has refused, claiming there is insufficient proof that it is under threat.
The Manx government is unable to make its case to Cites directly because the UK runs the Isle of Man's foreign relations. Manx ministers think Whitehall has no right to reject its request to put forward the shark proposal to other treaty nations.
Steve Rodan, a member of the House of Keys, the Manx Parliament, and the island's own department of the environment, said: "I'm very concerned that the UK government is choosing not to promote our interest."
He said provided the Manx request was reasonable, which it was, "the Government has a duty to do so".
Dr Simon Lister, director of the nationwide network of county Wildlife Trusts, said: "I'm furious . . . it seems outrageous that one or two officials in the Department of the Environment have taken it upon themselves to block this very worthwhile proposal."
Dr Lister wants the Secretary of State for the Environment, John Gummer, to take a direct interest.
The Worldwide Fund for Nature and the Manx government have pleaded with the DoE to think again. There are just five days left before nations have to submit their proposals for the next Cites treaty meeting, in Zimbabwe in June.
Labour's campaign co-ordinator on the environment, Joan Walley MP, has asked for an urgent meeting on the issue with Mr Gummer.
The DoE says its own scientific advisers believe there is insufficient evidence to back the proposal and not enough time to consult with other countries.
But Dr Sarah Fowler, a leading shark authority, said there had been several occasions around the world when over-fishing had caused collapses in basking- shark numbers, including one off Ireland's north-west coast in the 1950s. Sharks are slow breeders, with females taking decades to reach sexual maturity and bearing only a few young each year.
They used to be caught for their liver oil but now the main threat comes from rising demand for their fins. In Singapore, they have been fetching over pounds 200 a kilogram. The nearest place to Britain where they are caught deliberately in large numbers is Norway, where some whaling boats now harpoon them.
Several hundred of the sharks arrive in the waters around the Isle of Man each summer. They swim near the surface with mouths gaping wide. Each hour a water volume equal to a large swimming pool passes through their gills and the plankton is filtered out on "rakers" attached to them.
Ken Watterson, who runs the Manx basking-shark project and the sighting surveys, said: "In the last few years the numbers around here have gone from thousands to hundreds."
The proposal to Cites from the Manx government would not ban the international trade in shark products, but it would make treaty nations monitor it, and reduce it if it was shown to unsustainable.
The proposal would probably run into insurmountable opposition from South- East Asian countries, which do not want Cites covering fish stocks. But, say conservationists, merely getting it debated would be a start.