When the American defence secretary Donald Rumsfeld heard there was a paper circulating called How to Win in Iraq, he asked to see a copy. One could say charitably he merely wanted to stay abreast of the latest thinking. But what if, as many of his critics suspect, he is feeling stumped and has no idea how to win?

The primer, by Andrew Krepinevich — a former army officer and counter-insurgency expert — and published in this month’s issue of Foreign Affairs, has become a must-read in Washington and its author has since dropped in on the Pentagon, the CIA and vice-president Dick Cheney’s aides for briefings.

Indeed Krepinevich is so well-connected to some of the key players, from US army generals to Zalmay Khalilzad, America’s ambassador to Baghdad, that many believe his essay is the closest thing the Bush administration has to an actual road map.

For Krepinevich has said the unsayable: it will take the US at least a decade to build a secure Iraq, cost hundreds of billions of dollars, and mean heavier military casualties.

“People say the president has no plan, but Krepinevich is articulating it,” says John Pike of national security policy think tank GlobalSecurity.org. “The administration just can’t say it out loud.”

Krepinevich calls his proposals the “oil-spot” strategy: “It’s like an ink spot. When a fountain pen drops some ink and it hits the paper, it begins to expand.”

Basically, American forces should stop trying to kill as many insurgents as possible and concentrate on providing security and opportunity to the Iraqis in key “green” zones such as Baghdad and Mosul. “Then, over time, broaden the effort — hence the image of an expanding oil spot,” he says.

It would be slow, require the embedding of many US troops with Iraqi forces, and risk many American lives. Yet “my strategy is the best of a sorry lot. There is no way to win quickly or cheaply”, Krepinevich insists.

It is a particularly bleak time for defenders of the war. Last week’s stampede of pilgrims on a bridge over the Tigris, which left nearly 1,000 dead, added to the sense of despair in Iraq and America.

“This was obviously a huge tragedy, much greater than anything the insurgents have been able to achieve, but what it showed is that death and destruction have become all too common in Iraq,” Krepinevich says. “It’s a bit of a stretch to say this will be the straw that breaks the camel’s back, but there is increasing frustration with the Iraqi government’s inability to improve public safety and security.”

Americans are losing patience. Bush’s approval ratings are plummeting (he hit 40% in one poll last week) and that was before Hurricane Katrina wiped out New Orleans.

Last week, Francis Fukuyama, the author of The End of History, accused Bush of squandering the overwhelming mandate he received after 9/11. There was no good way out of Iraq that he could foresee; “everything to be regretted” about going in there in the first place.

Even so, he did not call for a pullout. “That would set off a chain of unfortunate events that would further damage American credibility around the world and ensure that it remains preoccupied with the Middle East, to the detriment of other important regions . . . for years to come.”

There is an emerging consensus that Rumsfeld should be sacked, not just from Democrats, but from influential neo-conservatives such as Bill Kristol, editor of the Weekly Standard magazine; Senator John McCain, the likely 2008 Republican presidential challenger, is also critical. With his loose talk about drawing down US forces — which was explicitly rebuked by Bush — Rumsfeld is increasingly isolated. “He’s like the crazy aunt in the attic who won’t shut up, ” said one Middle East expert.

But one salutary sacking does not amount to a strategy. Krepinevich, who would like Rumsfeld’s support, is tactful: it is more important to change course than personnel, he says. But he is almost as critical as Fukuyama about the conduct of the war to date.

Krepinevich was initially persuaded that Saddam Hussein was a threat (“based on the information we were given”). “If you had to do it over again, would you do it?” he muses. “Well, you don’t get that chance. I was surprised there was no Plan B.”

He assumed there must be one he simply hadn’t heard about. But last autumn he outlined some of his ideas to 30 generals, many of whom had served in Iraq. “I thought they would say, ‘You’re crazy, this is what we’re doing’, but they all sat there silently.” The problem is that, “Maybe we have forgotten how to win.”

Vietnam was so demoralising that the army’s doctrine became, “Go in big, win fast, leave in a hurry.” Under Bill Clinton it became axiomatic that you “needed an exit strategy before you got in”. The military was restructured accordingly and its ability to wage an effective counter- insurgent campaign was lost.

The most successful example of the “oil-spot” strategy, in his view, was the British campaign against Malayan insurgents in the 1950s. “You had Malays, Indians and Chinese, and no one group liked the other, and yet the British were able to wage a classic counter-insurgent strategy.”

Even so it took 12 years for the British to prevail and in Vietnam, his other source of inspiration, Krepinevich admits, “We didn’t even win.” Then, however, America also focused on killing insurgents at the expense of winning “hearts and minds”.

It is possible, he suggests, that America could operate the “oil-spot” strategy with fewer troops. “More of the same is more of the same,” Krepinevich says. “We tried that in Vietnam.”

The trouble with relying heavily on Iraqi forces, GlobalSecurity.org’s Pike points out, is that “any strong enough to suppress the insurgency would be strong enough to suppress the government”.

It could mean Iraq ends up back in the hands of a strongman, an outcome that was once seen as wholly undesirable. But there is a certain realpolitik re-emerging in some quarters in Washington. The country has become so fractured that “We would almost be lucky to end up with a strongman, given where we are now,” Krepinevich says.

Whatever strategy America pursues, this would not count as winning.