"I think it could be more than 10 million, I think, I hope," Farid Ayar, one of seven commissioners on the Electoral Commission, said on Saturday.
"I was thinking that maybe we could get around 11 million voters. But Iraqis are getting more used to going and voting now, so perhaps it was a little bit quieter ... and it was Ramadan," he said, referring to the Muslim fasting month.
If 10 million of the eligible 15.5 million voters cast ballots, that would give a turnout of about 65%, higher than the 58% recorded in January's election, the first held after Saddam Hussein's overthrow.
Sunni Arabs voted in surprisingly high numbers, many of them hoping to defeat it in an intense competition with Shia and Kurds.
The higher-than-forecast Sunni turnout made it possible the vote would be close - or even go the other way - and cast doubt on US hopes that the charter would succeed in moving Sunnis away from the ongoing violence.
Sunni Arabs voted in surprisingly
The bar to defeat the constitution is high: The opponents must get a two-thirds vote in any three of Iraq's 18 provinces. They are likely to reach that threshold in the vast Sunni heartland of al-Anbar province in the west.
They must win the other two from among the provinces of Salah al-Din, Ninevah or Diyala, north of Baghdad.
Each of those provinces has a Sunni Arab majority, but they also have significant Shia or Kurdish minorities.
Commissioner Ayar said voting had gone well, despite hiccups in some areas, particularly al-Anbar province, west of Baghdad, where armed men exchanged fire with US and Iraqi troops in the city of Ramadi.
Al-Anbar centres closed
"In Anbar, we couldn't open all the centres. There were 207 centres that were supposed to open there and I think we opened 144," he said.
Later he said about 5850 of the planned 6230 polling sites nationwide had opened.
"But the problems were not very big and we are very happy that we finished the process without hearing that anybody was killed in the streets."
At a news conference, the Electoral Commission officials said eight of Iraq's 18 provinces saw turnout above 66%.
Officials say eight of Iraq's 18
provinces saw turnout above 66%
In seven provinces, turnout was between 33% and 66%.
Two provinces indicated a turnout below 33%, the officials said. No data was available for al-Anbar.
Speaking to Aljazeera from the western town of al-Qaim, an Iraqi journalist, Falih Abd al-Karim reported: "There were no polling centres in the town so we could not participate in the process."
Because of the US military operation Iron Fist, residents of al-Qaim - estimated to be more than 150,000 - were unable to vote, Abd al-Karim said.
The journalist added that if residents had been able to vote, most had intended to vote against the constitution.
In the western town of Rawa, to the east of al-Qaim, a polling centre was open and attendance was high.
Abd al-Karim reported that most locals had voted against the constitution.
In Haditha, two centres opened but attendance was poor because of the intense presence of US and Iraqi forces, with Humvees and armed vehicles parked in front of each centre.
High turnout had been expected in the Shia Muslim south and the Kurdish north, and low turnout in Sunni Arab areas.
Ballot counting has already begun. Commissioner Ayar said he expected that the commission would be able to release partial results on Monday, and a final, but still uncertified, tally on 20 October.
"All of the monitors we talked to said that everything had gone well and that there were no serious problems"
Asked whether the election had been free and fair, Ayar said the international and United Nations' monitors had told electoral commission staff that they were satisfied.
"All of the monitors we talked to said that everything had gone well and that there were no serious problems," he said.
Ayar gave no clues about the results, but polling centre directors in some areas were more forthcoming.
At the Sajdat voting centre in Najaf province, the director said that of 3125 registered voters 2099 had cast their ballots.
All but 30 had voted yes - an approval of more than 98%.
In Miqdadiya, in Diyala province north of Baghdad, the head of Konoz polling centre said 2166 voters were registered, of whom only 366 turned up, 299 of them voting no and 67 yes.
In Yathrib, a Sunni mostly Arab town north of Baghdad, 3500 people voted, with 3497 of them voting no and just three yes.