Forensic experts in Iraq are working to identify a body thought to belong to one of five British men seized in Baghdad in May 2007.
The bodies of two were returned in June this year and the families of two others were told to expect the worst.
The third body is thought to be that of one of them - Alec MacLachlan, of south Wales, or Alan McMenemy, of Glasgow.
The UK foreign secretary said he believed the fifth man - IT consultant Peter Moore - is still alive.
Mr Moore, from Lincoln, was being guarded by the others.
By Frank Gardner, BBC security correspondent
It is likely this is the body of one of the two remaining unaccounted for security guards. This will not come as a complete surprise but is still deeply upsetting for the families.
Of the fifth man, Peter Moore, there is still no firm news. The Foreign Office is working on the assumption he is still alive.
When the last hostage comes back, then serious questions are going to be asked.
The demands of the kidnappers always centred around releasing militants from US custody. The Americans are gradually closing detention centres and handing over militants to the Iraqis who are, frankly, releasing them fairly promptly.
Whatever strategy has been employed has not so far produced results. That may not necessarily be anybody's fault on the British or even Iraqi side - just part of the grim business of hostage-taking in Iraq.
Diplomats say the identity of the body handed to the Iraqi authorities could be established within 24 hours and the body is expected to be flown back to the UK by the end of the week.
In a statement, Foreign Secretary David Miliband said it was not possible to "definitively confirm" either that the body was a hostage or to whom it belonged.
But he added: "My thoughts and those of all of my colleagues in government are, of course, with the families of the British men kidnapped in Iraq.
"We are determined to keep this period of uncertainty for the families to a minimum."
Mr Miliband said he would make a further announcement "in due course" once the body had been formally identified and the families of all the hostages had been informed.
He said: "I renew my call, on behalf of the British government and the British people, to those holding the hostages to return them to their loved ones."
Mr Brown's spokesman said: "The prime minister's thoughts are with their families at this extremely difficult time.
"He will leave no stone unturned in the government's efforts to secure the release of the remaining hostages."
Mr Moore's estranged father, Graeme Moore, 59, who was not a nominated next-of-kin and had not seen his son for several years, said his friends and family faced an anxious wait for news.
Brown - I feel for the families
"We won't be sleeping well tonight. We won't find out any more until they identify the body," he said.
"There's always that little bit of uncertainty but the information that has come out of Iraq suggested that they separated Peter from the others early on."
Mr Moore had been working for US management consultancy Bearingpoint in Iraq. The other men were security contractors employed to guard him.
The group was captured at Baghdad's Ministry of Finance in May 2007 by about 40 men disguised as Iraqi policemen.
The captors were understood to belong to an obscure militia known as Islamic Shia Resistance in Iraq, which has demanded the release of up to nine of their associates held in US military custody since early 2007.
The Foreign Office insists the British government has not been directly involved in negotiations and that the Iraqi authorities have been acting as lead negotiator.
Little is known about the captives because of a media blackout during a large period of their captivity.
It originally came on the instruction of the hostage-takers who said they did not want publicity.
The hostage crisis has been Britain's longest for nearly 20 years.
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