Under the plans, Britain will cut the number of its troops from the present 8 500 to 3 000 by the middle of next year. The US will reduce its forces in Iraq from about 176 000 today to 66 000 by early next year.
The plans, outlined in a document put to a Cabinet committee chaired by Tony Blair, were revealed as the insurgency in Iraq claimed up to 40 lives and wounded dozens from suicide bombs in Baghdad, the Sunni Triangle and along the Syrian border.
The surge of bombings followed a lull over the last week and took the toll of Iraqis killed since the new government took office on April 28 past the 1,500 mark.
Reid's memo, Options for Future UK Force Posture in Iraq, marked "Secret -- UK eyes only", was leaked to the Mail on Sunday. It confirms a Guardian report last week outlining the views of British military commanders.
Reid says any final decision on force levels would depend on the security situation in Iraq, the state of the Iraqi security forces, and "internal Iraqi pressure".
However, he reveals that the Pentagon is pushing for early and deep cuts in the number of American troops in Iraq, despite the concerns of US commanders in the field.
Reid emphasises the financial advantage of cuts in British force levels in Iraq but adds that they would have an impact on other countries with forces there, notably Australia and Japan.
Italy's Prime Minister, Silvio Berlusconi, confirmed on Friday that he would start pulling his country's troops out of Iraq within two months.
Reid says in the memo that the government has a "clear military aspiration" to hand the al-Muthanna and Maysan provinces over to Iraqi control in October 2005 and the other two UK-controlled provinces -- Basra and Dhi Qar -- next April.
"This should lead to a reduction in the total level of UK commitment in Iraq to around 3 000 personnel, ie, small scale ... and an estimated halving in costs of around =A31-billion per annum currently," the document says.
"Though it is not clear exactly when this reduction might manifest itself, it would not be before around mid-2006."
Reid said there was a "strong US military desire" for "significant" force reduction. According to the document, the US wants to hand control to Iraqi forces in 14 out of 18 provinces by early next year, allowing a reduction from 176 000 troops to 66 000.
But it says the Americans are divided over the plans. It says the Pentagon wants a "relatively bold reduction" while US commanders in Iraq want a more cautious approach.
The document says the Japanese would be reluctant to stay if force protection was provided solely by the Iraqis. The Australian position might also be "uncertain".
British military chiefs are also concerned about severe pressure on the armed forces when Britain takes control next May of Nato's international force in Afghanistan -- another country where the Pentagon wishes to cut US military commitment.
There are now only a few hundred British troops in Afghanistan. That could increase to 4 000 next year.
Reid drew up his memo last month. In a statement, he said it was just one of many papers setting out possible options facing Britain and the US in Iraq. He underlined the government's promise to stay in Iraq "as long as is needed".
Sunday's most lethal blast occurred at an Iraqi army recruitment centre in western Baghdad when a suicide bomber blew himself up in a line of recruits, killing 25 people and wounding 47.
In the northern city of Mosul, a suicide car bomber hit a police convoy, killing four and injuring three. In the northern city of Kirkuk, four civilians were killed and 15 wounded when a suicide car bomb exploded near the city's hospital and municipal headquarters. Several of the wounded were hospital employees.
Two suicide car bombers killed seven Iraqi customs officials along the Syrian border on Sunday, the US military said.
There were grim reports of the massacre of a Shia family in eastern Baghdad, eight of whom, including a two-year-old, were shot dead in their sleep. Relatives of the dead accused Sunni Arabs of being behind the killings.
Police also reported that the bloated and bullet-riddled body of the president of the Iraqi karate association, Ali Shakir, had been found floating in a river south of Baghdad. Shakir (38) had been abducted by gunmen in the town of Latifiyah on Thursday.
At the weekend, Major General William Webster, the US officer in charge of Baghdad, said the number of car bombs in the capital had been reduced to about one a day in July from about two daily in June.
- Guardian Unlimited =A9 Guardian Newspapers Limited 2005