There is no foreign country that matters more to Iran than Iraq (except perhaps the US). The puzzle is why the US imagines that Iranian involvement in Iraq will melt away if it protests angrily.

Yesterday’s skirmish, in which the US arrested eight Iranians in Baghdad and then let them go after consulting the Iraqi Government, was trivial and irrelevant to the broader clash between the two countries. However, it is another small sign that Iraq’s Shia-led Government is prepared to side with Tehran against the US, if only to avoid antagonism.

There is no reason – although Tehran may not need one – to connect the incident with President Bush’s speech the previous night, in which he declared that Iran’s nuclear ambitions put the region “under the shadow of a nuclear holocaust”. He added that “Iran has long been a source of trouble in the region”, that “the US is rallying friends and allies around the world to isolate the regime” and that “I have authorised our military commanders in Iraq to confront Tehran’s murderous activities”.

His speech, to military veterans in Nevada, came only hours after Mahmoud Ahmadinejad, the Iran President, said that the power of the US was collapsing rapidly in Iraq and that Tehran was ready to step in to help to fill the vacuum.

From the moment that Saddam Hussein fell four years ago, Iran has tried to exert its influence over the country. Iraq, with its Shia majority, is a natural ally for the almost entirely Shia Iran. All that is new is that Iran’s President is prepared to express his interest so openly. It is no surprise that he is unbothered by provoking the US; it matters more that he now appears indifferent to whether he offends Iraq’s Prime Minister, Nouri al-Maliki, and other Shia leaders in Iraq.

The US, and Britain in the south, have placed some hope on the Iraqis’ sense of national independence, and irritation at Iranian involvement, in keeping Iran at bay. Those are real factors, but they have been matched by the effort and money that Iran has devoted to building ties, particularly in the south.

The US’s best hope of curbing Iranian influence remains Iraqis themselves, but it is a fragile one. For the local militias and factions jostling for the upper hand, Iranian support is potentially decisive. The US might better focus its efforts on the nuclear question, where its call for international support, and for tighter sanctions, has had some success. This week Nicolas Sarkozy, France’s new President, said that failure to pursue tougher sanctions would present the world with “an alternative that I can only call catastrophic: an Iranian bomb or the bombing of Iran”. Tehran has insisted that its intentions in acquiring nuclear technology are peaceful, but its 20-year record of deception, and continued obstruction of United Nations inspectors, has left the world disbelieving. For all its defiance in edging forward with its work, it has appeared sensitive to the penalties imposed by the UN Security Council. That has given the US, Britain and France, those calling for tighter sanctions, some hope of a deterrent effect.

Britain has differed from the US in being prepared to talk to Iran about Afghanistan and their shared aim of stopping the opium trade.

Bush’s speech raised the stakes by uniting all Iran’s offences into one overarching justification for its hostility. That may play a useful part in the diplomatic mixture, by lobbing an undefined threat of aggression in Iran’s direction. But, so far, the talks which have worked best have separated the points of dispute – from Afghanistan, where Iran is helpful to the West, to the nuclear front, where it appears, just possibly, open to persuasion, and to Iraq, where its interests lie too deep for it ever to be on the US side.

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