Negotiators for Iraq's Shia majority say a deal has been agreed on a final draft for the new constitution.
Sunnis in Baquba demonstrated against constitution plans
They say the text will be put to the Iraqi parliament for approval within the next two days.
But politicians for the minority Sunni Arabs flatly contradicted the Shia claim, saying there was no agreement despite talks late into the night.
Meanwhile, the US military has announced the release of 1,000 inmates from the notorious Abu Ghraib prison.
US officials have refused to say whether the decision to let the prisoners go is related to the talks over the constitution.
However the BBC's Jon Brain in Baghdad says that the release, which follows Sunni appeals, can only be seen as a positive move in the current climate.
The chairman of Iraq's constitution committee said on Friday that the final text of the constitution would go before the National Assembly in the next couple of days.
"There has been an agreement on the differences including the federalism issue," Sheik Humam Hammoudi, a Shia, said.
15 August deadline extended twice
National referendum on constitution by mid-October
Full government elections by mid-December
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"This will give guarantees for the Sunnis."
But top Sunni negotiator, Saleh al-Mutlaq, told al-Jazeera television that compromises offered by the Shias were "still far from what we need".
And he called on Iraqi people to reject the constitution in October.
"The Iraqi people have to give their word now and reject the constitution because this constitution is the beginning of the division of the country and the beginning of creating disturbance in the country."
Iraq's Vice-President Ghazi al-Yawer, a Sunni, has defended Sunni objections to the constitution.
"So far, it seems that the content of the draft constitution is somewhat remote from the aspirations of all segments of the Iraqi people," he said.
The outstanding issues from the Shia-Kurdish draft submitted last Monday included:
- federalism, and the way to form [federal] regions
- the terminology used in eradicating the influence of the former Baath regime - whether to use the term Baath party or Saddam's Baath
- structuring of authority between the presidency, parliament and the government.
Sunnis have expressed concerns that allowing for federalism may lead to the creation of an autonomous Shia area in southern Iraq - like the Kurdish north but under Iran's influence.
The Sunnis fear greater autonomy for the Kurdish north and Shia south could compromise their share of revenues from those oil-rich regions.
The original deadline last week was postponed twice - giving negotiators 10 extra days to reach a deal.
Shias and Kurds could approve the document in parliament without Sunni backing.
Sunni leaders fear federalism will split the country up
But the insurgency that still plagues Iraq has its roots in the Sunni heartlands and the constitution is supposed to be part of the process of winning the Sunni community round, the BBC's Baghdad correspondent Mike Wooldridge says.
It has emerged that US President George Bush phoned a Shia leader earlier this week urging him to seek consensus.
Mr Bush spoke to Abdul Aziz al-Hakim and "asked him to be more flexible with regard to Sunni demands," sources close to the ruling Shia alliance told Reuters.
Meanwhile, thousands of Sunni Muslims demonstrated in the Iraqi city of Baquba to protest against the constitution being debated in Baghdad.
Some danced and sang chants glorying Saddam Hussein, the deposed Iraqi leader who held onto power through a series of bloody crackdowns on the country's Shia and Kurdish communities.