But there are no signs yet that the new deadline of 22 August will be any easier for the drafting committee to meet than the original deadline.
That is because in the run-up to yesterday's deadline, even some areas of previous agreement fell apart in the heated bargaining.
Observers say the key disagreements are still over how much Islam should be a source for Iraqi law, and over demands from Shi'a religious parties to set up an autonomous Shi'a region in southern Iraq.
But now disagreements have also opened up over the division of Iraq's oil wealth, over whether Kurdish-administered northern Iraq has the right to secede if it wants to, and whether the senior Shi'a religious leadership, the Marjariya, should be declared independent of the Iraqi government.
"Yesterday, when the leaders went into the meetings, some of the participants said that they went in with a certain number of issues of disagreement, but they came out with more issues which were not settled," said Kamran al-Karadaghi, spokesman for Iraqi President Jalal Talabani.
Al-Karadaghi also said that the constitutional committee is trying to bridge differences between two draft versions of the charter -- one presented by the religious parties, the other reflecting more secular values.
"They have actually two texts, maybe, of a draft constitution written in different ways," al-Karadaghi said. "There is a more modern constitution, and another [in a] language with a lot of religious flavor, I would say. So, this is why they really couldn't reach an agreement. And they spent a lot of time, for example, discussing the preamble which was, again, two different languages, and also, of course, the other issues like, still, the resources, oil and gas, and federalism."
In face of the continuing disputes, Iraqi and U.S. leaders continue to express optimism that an agreement can be reached with another week's work.
Prime Minister Ibrahim al-Ja'fari told reporters in Baghdad after yesterday's vote to extend the deadline that "I think next week will be enough."
But some of participants in the negotiations are more skeptical.
"The differences, frankly, are huge, and it will take strong national will on all sides to reach a consensus," Reuters quoted Saleh al-Mutlaq, spokesman for the Sunni Arab umbrella group Iraqi National Dialogue, as saying. "The political parties should renounce their differences, and should reduce their differences, and think about what the Iraqi people want. The Iraqi people do not want division and do not want federalism."
Sunni Arab representatives on the constitutional committee have clashed particularly with the Shi'a religious parties over their demands for an autonomous region composed of nine Shi'a-majority provinces -- that is, half of Iraq's provinces.
The Sunni Arabs are reported to fear that Kurdish and Shi'a autonomous authorities might hoard Iraq's oil wealth, leaving the Sunni center of the country -- which has no oil fields -- impoverished.
And the Sunni Arab leaders are said to be angered by threats by some Shi'a religious party leaders to use their domination of the National Assembly to push through a draft constitution even if they can't get a full consensus in the committee.
The National Assembly must approve the draft constitution before it is submitted to a national referendum scheduled for October. Sunni Arabs are underrepresented in the assembly after they largely boycotted the January parliamentary election.
Some Sunni leaders have said that if the additional week does not produce a breakthrough, they will push for dissolving the National Assembly and holding a new election.
That would be in accordance with the Transitional Administrative Law approved by the U.S.-appointed Iraqi Governing Council a year ago. The law requires the dissolution of the current government if it cannot produce a draft constitution – something considered a key milestone along Iraq's path to democracy.
Al-Karadaghi said that the National Assembly voted yesterday to give the drafting committee only a week more because any longer delay would make it difficult to adequately prepare for the October constitutional referendum.
"This draft should be prepared, printed, and published [in time to] give two months for the people to get familiar with the draft constitution," al-Karadaghi said. "The idea is to distribute a copy with every [food] ration card so that more people will see the draft. It would be published in the newspaper, there would be a public debate, and then on 15 October there must be a referendum."
Under the Transitional Administrative Law, Iraq is due to reach one more milestone this year: a general election for a constitutional government in December.
But whether Iraq can hold that poll on schedule will only become clearer next week, as the constitutional committee either meets its deadline or again falls victim to its continuing disputes.