Sunnis said they would refuse to accept a charter that included federalism.
US Ambassador Zalmay Khalilzad said he was convinced that the Iraqis would resolve their differences and finish the document. Some politicians suggested the draft could be presented to the Shia and Kurdish-dominated parliament on Monday over Sunni Arab objections.
Despite broad agreement between Shia and Kurds, a constitution unacceptable to Sunni Arab leaders would further alienate the community, which has been undercutting the US goal of using the political process to take the steam out of the Sunni-dominated armed opposition.
The Iraqi parliament set a meeting for 6pm (1400 GMT) on Monday to allow as much time as possible to resolve the differences. A Sunni official said his group objected to other issues apart from federalism, including a Shia-Kurdish deal to grant special status to the Shia clerical hierarchy.
Last-minute talks focused on issues such as federalism and the role of Islam - the very obstacles that have blocked progress on the draft for weeks. Sunni Arabs oppose federalism, fearing it will lead to the breakup of the country.
Sunni Arab politicians met on Sunday with President Jalal Talabani and Massoud Barzani, leaders of the two major Kurdish parties and leading proponents of federalism, and asked that the issue be left out of the draft.
"We made a proposal to transfer federalism and the process of forming federal regions to the next National Assembly," Sunni politician Kamal Hamdoun said.
Iraqi women calling to include
Islam in the new constitution
"Legislation could be drafted on these two matters and a referendum could be held on them."
Hamdoun said the Sunnis received no response to their proposal, which the Kurds have rejected in the past.
Hamdoun said Sunnis also objected to other issues accepted by the Shia and Kurds, including recognition of the Kurdish language, dual citizenship and the role of the Shia religious leadership.
"I don't think that the draft will be handed tomorrow if they are serious about the unanimity with us," Hamdoun said. "If there are points that we do not agree on, we will not sign any draft. We expect anything could happen tomorrow."
In northern Iraq, the Kurds organised rallies on Sunday in Irbil, Sulaimaniyah and other Kurdish-dominated cities of the north to press their demands for self-rule.
Waving Kurdish green, red, white and yellow flags, the demonstrators carried banners reading: "In the new constitution, Kurds should be given the right of self determination."
Since Shia and Kurds have agreed on most remaining issues, Kurdish lawmaker Mahmoud Othman said the drafting committee would present the draft to parliament on Monday even if the Sunni Arabs object.
Iraqi women are calling for
equality in the constitution
Parliament could approve the draft by a simple majority, and the Shia and Kurds together hold 221 of the 275 seats. However, that risks a Sunni backlash which could scuttle the constitution in the 15 October referendum.
A 'no' vote
If two-thirds of the voters in at least three of the 18 provinces vote 'no', the constitution will be defeated.
Sunnis form a majority in at least four provinces, and Sunni clerics have urged their followers to vote against any constitution which could lead to the breakup of the country.
With the Sunnis standing fast, Shia legislator Jawad al-Maliki, a member of al-Jaafari's Dawa Party, said on Sunday that if politicians could not reach agreement, "we might amend the interim constitution and extend the deadline by a minimum of two weeks" to allow time to win over the Sunni Arabs.
That would require approval of two-thirds of parliament and the president and his two deputies. However, the United States is anxious to keep to the schedule and has ratcheted up pressure on the Iraqis to meet the Monday deadline.
Kurdish parliament backs Kurdish
autonomy in northern Iraq
A survey conducted by Iraq's constitution drafting committee showed that 69% of respondents support full rights for women - as long as the freedoms do not contradict Islam.
The survey, released on Sunday, was conducted by a subcommittee of Iraq's constitutional committee that is drafting a new charter for the country. The group was trying to gauge public opinions on key topics that were being addressed in the charter.
More than 156,000 people submitted responses to a multiple-choice questionnaire that was distributed across the country.
Participants handed the forms into 1000 boxes across the country, according to Adnan Mohammed Hassan, head of the dialogue committee that directed the survey.
On the topic of women's rights, 12% of respondents said women should have the same rights as men. Some secular-minded women fear a loss of rights if conservative clerics heavily influence the new constitution.