The attack happened when RAF Tornados based in Kuwait and Saudi Arabia were patrolling the "no-fly" zone over southern Iraq imposed after the Gulf War.
Iraq says both the southern and northern no-fly zones are illegitimate and that it will shoot down aircraft enforcing them.
United States and British aircraft were patrolling the southern zone at about 8.30am local time yesterday when an Iraqi battery fired between six and eight missiles from a site south-west of Talil, in southern Iraq. Two US F-16 bombers responded and fired two anti-radar missiles and a number of precision-guided bombs on to the site. All the allied aircraft involved in the incident returned safely to their bases in the Gulf.
Iraq, however, said it had almost certainly shot down a US or British plane over the no-fly zone. "Our brave air defences have fired ground- to-air missiles against the formation of hostile planes, forcing them to flee after one of the planes was almost certainly shot down," a military spokesman said.
The Secretary of State for Defence, George Robertson, said the allies would not be "intimidated" from policing the no-fly zones by the Iraqi leader, President Saddam Hussein. "They [the no-fly zones] are humanitarian in purpose and we intend to maintain them," he told BBC Radio 4. "What we are seeing now is a show of defiance, arrogance and essentially of weakness on behalf of Saddam."
The incident was the second this week involving missiles, and follows the attack on US aircraft on Monday in the northern no-fly zone, near Mosul.
The US and Britain hit hard at Iraq's air defences during Operation Desert Fox, a 70-hour air campaign earlier this month. But they admitted at the time that Iraq's surface-to-air missiles were highly mobile and many were missed.
Despite the threat to its aircraft, the US also said the patrols would continue. "This is a key element of our containment policy to prevent Saddam Hussein from using his aircraft to threaten his own people and his neighbours," said a spokesman for the National Security Council. "We will continue to vigorously enforce it and our aircraft will take the necessary precautions to carry out their mission and defend themselves."
Soon after news broke of the clash, President Bill Clinton spoke about Iraq to the Russian President, Boris Yeltsin. Russia has been critical of US policy towards Baghdad although both agreed that despite their differences "it's important to continue to build the US-Russian relationship", the White House said.
But in a sign that the clashes are worrying Iraq's neighbours and America's allies, Turkey said it wanted the issue to be handled delicately. "This is a sensitive situation for Turkey," said the Defence Minister, Ismet Sezgin. The allied air forces depend on bases in Turkey, Kuwait and Saudi Arabia.