The UK plans to withdraw its 4,100 troops by the end of July 2009
A compromise has been reached to let Iraq's parliament vote on a resolution allowing non-US forces to remain in Iraq beyond the end of 2008, MPs say.
On Saturday, they rejected a bill which gave the 6,000 troops from the UK and other states a legal basis to stay once the UN mandate expires on 31 December.
A resolution would allow the government to sign deals with each country.
The 140,000 US troops are allowed to remain until the end of 2011 under a separate status of forces agreement.
Earlier, UK Defence Secretary John Hutton described Saturday's vote as "a minor hiccup", but said he was confident a deal would be reached.
The UK has already said it plans to withdraw by the end of July.
A day after the Iraqi cabinet's draft bill was rejected, members of the Council of Representatives said the main parties had agreed to approve a resolution which should allow non-US troops to stay on until the end of July 2009.
Ali al-Adeeb, a member of the governing United Iraqi Alliance told reporters the parliament would vote on the resolution on Monday.
The BBC's Caroline Wyatt in Baghdad says a parliamentary resolution should only need a simple majority to pass, which the governing parties are optimistic about achieving.
A resolution would authorise the Iraqi government to sign separate bilateral deals with each nation which would give them the legal protection their troops need to remain beyond the end of this month, our correspondent says.
The move is a way of getting around opposition to a law, she adds.
Some MPs said they had rejected the draft because its terms were not as strict as the pact governing US forces agreed this month.
Others, loyal to the radical Shia cleric, Moqtada Sadr, said they wanted the foreign troops to leave Iraq when the UN mandate ends.
NON-US FORCES IN IRAQ
UK - 4,100
Australia - 1,000
Romania - 500
El Salvador - 200
Estonia - 40
Earlier in the week, UK Prime Minister Gordon Brown announced that the UK planned to withdraw its troops from southern Iraq by the end of July 2009, as envisaged by the draft law.
Military operations will end by 31 May and the remaining 4,100 service personnel will leave within two months. Several hundred trainers will remain, some working with the Iraqi navy.
On Sunday, the British defence secretary said it would be "very serious" if an agreement was not finalised before 31 December, but added that there were "contingency plans" in place in case should that happen.
Apart from the US and UK, the only countries continuing to provide troops for Multi-National Force - Iraq (MNF-I) are Australia, El Salvador, Estonia and Romania.
South Korean troops ended their mission in Iraq on Friday, joining their Japanese counterparts, who pulled out a day earlier.