Rumours that the formula could become a global standard have especially rattled Germany, Japan and South Korea, which host thousands of forces, and US officials have mentioned the demand to at least one country in a formal negotiation setting, said people familiar with the matter.

National Security Council spokesman Garrett Marquis said the Trump administration “is committed to getting the best deal for the American people” but would not comment “on any ongoing deliberations regarding specific ideas”.

Trump has long complained that US and Nato allies freeload on US military protection, but the cost-plus-50 formula has only gained traction in recent months, said current and former US officials, who like others spoke on the condition of anonymity to discuss sensitive negotiations.

It is not a formal proposal or policy but serves as a kind of “maximum billing” option designed in part to draw attention to an issue that speaks to Trump’s demand that allies shoulder more of the burden of their own defence, a senior administration official said.

One of the first US allies to confront the Trump administration’s hardball tactics was South Korea, which last month agreed to pay US$925 million for hosting 28,500 American troops. That was an 8.2 percent increase from the previous year’s payment and about half the total costs. South Korean officials preferred a five-year agreement, but the deal covers only one, meaning they could face pressure to meet Trump’s cost-plus-50 demand next year.

A US military official said US Forces Korea had been “sweating” the signing of a new agreement for months.

There are numerous burden-sharing ideas floating around, and Trump has not settled on any one, officials said.

Although it may be a red herring, the phrase “cost plus 50” has appeared on informal lists of options, one official said. But it is not clear what Trump advisers mean by “cost,” whether it’s the entire budget to run a base and pay US armed forces or some part of that.

US allies hosting permanent American military installations pay for a portion of costs in various ways. Japan and South Korea make cash contributions, while Germany supports the US troop presence through in-kind contributions such as land, infrastructure and construction, in addition to foregone customs duties and taxes.

Trump has called that “in-kind” contribution insufficient, a senior US diplomat said.

For decades, leading foreign policy figures in both parties have urged US allies to take on greater responsibility for their security, but even staunch advocates of burden-sharing have questioned Trump’s approach.

“Trump is correct in wanting US allies to bear more responsibility for collective defence, but demanding protection money from them is the wrong way to do it,” said Stephen Walt, a scholar of international relations at Harvard University. “Our armed forces are not mercenaries, and we shouldn’t send US troops into harm’s way just because another country is paying us.”

The cost-plus-50 idea would probably not be presented as a blanket demand to all allies, even if Trump ended up signing off on it, several people familiar with elements of the discussion said. Many of his top aides oppose the formula and have succeeded in the past in bringing him down from the maximalist approach, the people said.

The existence of Trump’s formula was first reported by Bloomberg News.

Critics of US bases around the world say the bases are costly, stoke tensions with adversaries and have unintended consequences. The Pentagon counters that its 54,000 troops in Japan and presence in South Korea allow it to project power and deter North Korea and China.

In Germany, where the Pentagon has more than 33,000 troops, the US Army announced last year that it could add 1,500 more by 2020 in “a display of our continued commitment to Nato and our collective resolve to support European security.”

Trump’s idea has been rumoured in European capitals for months, though senior European diplomats said they knew of no formal presentations or threat from the White House. Such a proposal appears aimed principally at Germany, the subject of frequent Trump complaints about Nato defence spending and what he says is an unfair German reliance on American forces for its defence.

Trump does not accept the argument that US forces in Germany are a strategic asset for the United States and maybe an overall cost savings because they help facilitate US military actions in the Middle East and Africa as well as across the European continent, former US officials said.

That disconnect predates the discussion of billing Germany for the cost of basing forces there, and some former advisers had hoped they could steer Trump toward a wider view of what the United States gains from the arrangement. American lives that might have otherwise been lost on the battlefields of Afghanistan, Iraq and elsewhere, for example, are often saved at Landstuhl military hospital in Germany.

“When he says, ‘Thirty thousand American forces are there protecting Germany,’ that is a completely inaccurate explanation of what American forces in Germany are there for,” retired Lieutenant General Ben Hodges III, said in an interview in the fall as Trump’s rhetoric on the issue heated up.

Hodges was addressing the president’s complaints about the number of US forces in Germany – more than 30,000 – and threats to downsize or relocate forces, not the specific idea of billing Germany.

The benefit to the United States cannot be measured in the transactional ways Trump frames it, said Hodges, who served as commanding general of the US Army in Europe.

“Like with our base in Ramstein, this is a platform for power projections in the Middle East, Africa, Russia.”

Emma Ashford, a scholar at the libertarian Cato Institute, agrees with Trump that the US military is overextended but said his latest gambit is the wrong tactic.

“The solution to America’s unbalanced commitment to rich allies is to gradually shift the burden to them and remove the troops,” she said. “Not to keep American troops there and charge for them like they’re mercenaries.”

The discussion comes as allies prepare for the annual summer summit, where Trump has twice berated German Chancellor Angela Merkel over her country’s defence contributions. Trump routinely misstates the Nato funding arrangement and defence spending targets, but Germany acknowledges that it has not met the threshold goal of spending 2 percent of gross domestic product on defence.

Trump could undermine the effort to increase European Nato defence spending if he starts demanding bilateral payments, said Jeffrey Rathke, president of the American Institute for Contemporary German Studies at Johns Hopkins University.

“The United States, including under the Trump administration, has had a lot of success in persuading Germany and other Nato allies that they need to contribute more to their own defence,” Rathke said. “That is possible because the spending is directed at a common Nato objective, and that is collective defence,” which is more politically palatable in western Europe.