So far, voters in only one province have rejected the draft charter by a two-thirds majority, according to the figures released by the electoral commission.
But two provinces with large Sunni Arab populations, among whom opposition to the text runs high, are among those that have yet to return their results.
Under the rules for the 15 October referendum, the constitution fails if it is rejected by a two-third majority in any three of the 18 provinces and elections to a new parliament must be held.
Province rejects text
In Salah al-Din province, location of Saddam Hussein's hometown, 81.5% of voters rejected the text.
The commission said the remaining five provinces - mainly Sunni al-Anbar and Nineveh, mainly Kurdish Arbil and mainly Shia Basra and Babel - are expected to release their results in the next few days.
The murder of an Iraqi lawyer
sparked calls for more security
"The results from these governorates and audit reports will be presented as part of the final preliminary results scheduled for release early this week," the commission said.
"This audit is scheduled to be completed in the coming days at which point the IECI will release a complete set of provisional
US troops killed
Elsewhere in Iraq, the US military announced on Saturday the deaths of four more soldiers.
Three US marines and a soldier were reported killed on Friday, the US military said.
Two marines "were killed in action while conducting combat operations against the enemy" when their vehicle was hit by a makeshift bomb near al-Amariyah, west of Baghdad on Friday.
The number of US soldiers killed
in Iraq is nearing 2000
Another marine was killed "while conducting combat operations
... when he was hit by an explosion" the same day near the town of Haqlaniyah, which is also in the western al-Anbar province, a statement read.
Separately a US soldier in Baghdad died from a "non-hostile" shooting.
The latest deaths brought to 1991 the number of US military personnel killed in Iraq since the US-led invasion began according to an AFP tally based on Pentagon figures.
The government promised on Saturday to step up security for defence lawyers in Saddam Hussein's trial after one was kidnapped and killed by men wearing security forces' uniforms, but the lawyers rejected an offer of Iraqi guards, suspicious of Iraq's Interior Ministry.
Saadoun al-Janabi was one of 13 lawyers who appeared at the first session of Saddam's trial on Wednesday representing the ousted dictator and seven top officials from his Baathist government.
"Everyone knows there are elements in the Interior Ministry that assassinate Iraqis"
Khamees Hamid al-Ubaidi,
lawyer of Saddam Hussein
"We have decided to take some measures to protect the lawyers," Deputy Interior Minister General Hussein Ali Kamal told The Associated Press on Saturday, though he refused to
But one of Saddam's two lawyers said the entire defence on Friday night had rejected an offer of guards from the Iraqi Interior Ministry. He suggested they want protection by American forces.
'Lack of trust'
Khamees Hamid al-Ubaidi pointed to frequent Sunni Arab accusations that Interior Ministry forces or Shia militias linked to the government have carried out killings of Sunni Arabs.
"We refused because of our lack of trust in the Iraqi security agencies," al-Ubaidi said. "Everyone knows there are elements in the Interior Ministry that assassinate Iraqis."
He said the defence lawyers were talking with the Americans about protection and want US officials to carry out the investigation into al-Janabi's slaying.