Sharp differences remain on federalism, the role of Islam and sharing of oil wealth, some of the key planks of the first post-Saddam Hussein charter which is due to be put to parliament on Monday after a 15 August deadline was missed.
"We have a problem here... there is one group who wants a 21st century constitution and there is another group who wants a seventh century constitution," said one source closely involved in the negotiations.
"Unfortunately, America is looking at both the groups with the same eye. They just want the draft to be ready on time."
The Kurds, who want their de facto autonomous northern region to include the oil centre of Kirkuk, have been demanding first rights to the oil produced there.
Last week, negotiators proposed one formula for distributing Iraq's vast oil wealth whereby each oil-producing region would take a small percentage for itself, with the rest transferred to Baghdad for national distribution.
An exact arrangement is still to be worked out, and the Kurds are pushing for maximum gains.
Iraq's constitution is seen as key to the country's political transition and possible early withdrawal of US-led troops.
Iraqis will vote on the constitution
in October ahead of new elections
It is due to go to a referendum in October ahead of new elections in December.
Sources close to the negotiations said US ambassador Zalmay Khalilzad, who attended meetings until late on Saturday, has asked the Kurds to soften their stand on oil as well as their demand for self-determination.
"The US is pressuring the Kurds to give up these two demands," said one source.
Kurdish leaders on Saturday offered to compromise on self-determination.
They had been keen for language to be included in the charter giving them the right to self-determination, which would effectively allow them to secede from Iraq at some point in the future.
Zalmay Khalilzad has asked Kurds
to soften their stand, say sources
The United States on Saturday dropped its opposition to enshrining Islam as "the" main source of legislation and not just "a" main source - a move aimed at pleasing the majority Shias.
Washington is determined to see the date met after the first deadline was missed last Monday, fearing that any delay in the political process will benefit Sunni Arab fighters opposed to the government.
The Kurds have rejected moves to make Islam as "the" main source of law, saying it would harm women's rights and Iraq's secular tradition.
"We will oppose this as much as we can," Kurdish constitution committee member Mahmud Otham said.
"We have a problem here... there is one group who wants a 21st century constitution and there is another group who wants a seventh century constitution"
Source involved in the negotiations
The role of Islam has proved a heavily divisive issue, with leaders of the Shia majority insisting religion be considered the main legal foundation, and that clerics be given political roles.
One Western official said "no one is looking to establish an Islamic state. The intent is to ensure that Islam is respected in addition to other established rights."
Observers speculate that the Shias and Kurds, who enjoy a majority in parliament, may forge a compromise over the heads of Sunni negotiators.
"We have been sidelined... we have met the leaders only twice since the new deadline," said Saleh al-Mutlaq, a constitution committee member from the former Sunni Arab elite which largely boycotted January's landmark elections.
An al-Qaida-linked group has
warned Iraqis to boycott poll
"They (Shia and Kurds) will prepare a draft and ask us to sign. If we do it, we will be blamed by our people and if we do not the leaders will blame us for obstructing the political process. This is unfair."
Mutlaq had warned that the Sunnis would defeat the constitution at the during the mid-October referendum if they are ignored.
Under Iraq's interim law, the charter will fail if two-thirds of voters in any three provinces reject it in the referendum.
Sunni Arabs form a majority in al-Anbar, Ninevah and Salaheddin provinces.
Ansar al-Sunna, a group linked to al-Qaida, warned Sunnis to boycott the referendum.
"A constitution is for illegitimate states," it said in an internet statement on Sunday.
"Anyone who obeys a law other than God's law is a miscreant."
With August one of the deadliest months for American troops since the March 2003 invasion, US President George Bush defended the Iraq war again in his weekly radio address on Saturday.
"We must finish the task that our troops have given their lives for and honour their sacrifice by completing their mission," said Bush, whose approval ratings have slipped to some of the lowest levels of his presidency.