"If the text is not handed to the national assembly by the deadline ... one choice is to ask for another one-week extension or the other is to dissolve the parliament," said Leith Kubba, spokesman for Prime Minister Ibrahim al-Jaafari.
After missing the original 15 August deadline to submit the first post-Saddam Hussein charter to parliament, Iraq's leaders secured an extension allowing them to postpone an agreement until 22 August.
But sharp differences remained on Sunday on issues including a federal structure for Iraq, the role of Islam and the sharing of national oil wealth, raising the prospect of another parliamentary vote to extend the deadline.
Aljazeera reported on Sunday that a Jawad al-Maliki, a member of al-Jaafari's Iraqi Daawa (Call) Party, said the issue of federalism would "finally" be settled and that the final version of a draft constitution would be submitted by Sunday.
However, another Iraqi source close to the negotiations who did not wish to be named, said the constitution wrangling was because "there is one group who wants a 21st century constitution and there is another group who wants a seventh century constitution".
"Unfortunately, America is looking at both the groups with the same eye. They just want the draft to be ready on time," he added.
Washington sees the charter as key to keeping Iraq's political transition on track in the face of a deadly uprising threatening to engulf the country.
Negotiators reported that US officials were pressing Kurds to give up their ambitions of self-determination and water down demands on oil ownership in a bid to reach a deal with the majority Shia.
"There is one group who wants a 21st century constitution and there is another group who wants a seventh century constitution"
Others said US ambassador Zalmay Khalilzad had personally persuaded Kurdish groups to soften their stance.
Kurdish leaders on Saturday offered a compromise on self-determination, a demand which would in effect give their de facto autonomous northern region the chance to secede from Iraq at a later date.
But Kurdish ambitions to have the oil centre of Kirkuk included within their territory and to seek a degree of control over the region's oil reserves could be more difficult to assuage.
Installing Islam as the country's main source of legislation and allowing clerics a political role, as demanded by the conservative Shia bloc, are also stumbling blocks.
The US on Saturday dropped its opposition to enshrining Islam as "the" main source of legislation and not just "a" main source in an effort to please the majority Shia.
But the secular Kurds strongly oppose the move, arguing that it contravenes women's rights and the country's secularist traditions.
"We will oppose this as much as we can," Kurdish negotiator Mahmud Otham said.
Observers speculate that the Shia and Kurds, who enjoy a majority in parliament, could forge a compromise deal - but this is likely to be over the heads of Sunni negotiators.
Iraq's Sunnis reject any notion of federalism - the minimum Kurdish demand - although their negotiating position is weak since they hold few parliamentary seats after largely boycotting January's elections.
But the Sunni members have warned that if they are sidelined in any constitution deal, the charter could well be defeated in the scheduled mid-October referendum.
Under Iraq's interim law, the charter will fail if two-thirds of voters in any three provinces reject it.
Sunni Arabs form a majority in at least three provinces: al-Anbar, Ninevah and Salah al-din.
Also on Sunday, Baghdad called on neighbouring Jordan to extradite members of Saddam's former government, claiming they were fomenting "terrorism" in Iraq from the Hashemite kingdom.
Saddam could face death after
Baghdad defended the penalty
The United Arab Emirates and Syria were also included in a list of countries Baghdad said were being used by groups backing fighters in Iraq.
More than 280 foreigners are currently being held by Iraqi authorities on suspicion of "terrorism", Baghdad said, including an unnamed Briton.
The Iraqi government also blasted Jordan for allowing Saddam's family to conduct "hostile" political activities there.
Saddam, currently in US custody awaiting trial for crimes against humanity, could face execution if found guilty after Baghdad on Sunday issued a strong defence of its decision to reinstate the death penalty.
The UN had protested against the planned executions of three convicted felons, but Baghdad said there was "consensus" in Iraq on capital punishment - banned since the 2003 US-led invasion.