Iraq's President Jalal Talabani has said he thinks a deal can be reached on the new draft constitution on Sunday, one day ahead of Monday's deadline.
Some movement is reported on Shia Iraqis' religious demands
Mr Talabani said Iraqi politicians had resolved many differences - but talks continue on the key issues of power sharing and Islam's role in law making.
The US urges compromise but observers say some issues may have to be shelved if a consensus is to be reached.
The draft constitution is due to be put to Iraq's parliament by Monday evening.
President Jalal Talabani expressed hope a deal would be ready on Sunday. "Many of the contentious issues have already been resolved," he said.
The focus of the talks, he added, was the role of religion and the question of "federalism for the south", where the majority Shia Muslim community predominate.
One negotiator said the US ambassador, Zalmay Khalilzad, had come up with a text of his own to try to bridge the gaps.
The BBC's Caroline Hawley in Baghdad says progress has been reported in some areas but a lot of tough bargaining lies ahead, particularly on the issue of federalism.
The draft is supposed to be handed to parliament for scrutiny by 15 August, ahead of a nationwide referendum planned for October before a general election scheduled for December.
New constitution to be completed by 15 August
National referendum on constitution by mid-October
Full government elections by mid-December
The sharing and distribution of oil revenues would be the most significant among the issues on which at least tentative agreement is now being suggested in some quarters, the BBC's Mike Wooldridge reports.
Also, Kurdish and Shia groups have dropped demands that the country should be called federal or Islamic, and agreed that the official name should be the Republic of Iraq.
But the future relationship between central government and Iraq's various regions remains contentious.
While there is general agreement that the Kurds should be allowed to keep self-rule in the north, Sunni Muslims have been incensed by a call from the leading Shia politician, Abdul Aziz al-Hakim, for an autonomous region in the oil-rich south.
At Friday prayers, Sunni clerics condemned federalism.
Our correspondent says that if no breakthrough is achieved, the politicians may approve a constitution that essentially shelves federalism and maybe also other disputed issues that have to do with Iraq's identity.
Nicholas Haysem, a UN constitutional advisor in Iraq, said such an approach would not necessarily mean greater problems were stored up for the future.
US President George W Bush said on Saturday he was confident Iraq would get a new constitution later this year, despite difficulties Iraqi politicians are having in talks on the document.
He said the constitution would be a critical step on the path to Iraqis themselves taking control of their country and its defence - but he said again there would be no premature withdrawal of US troops.