The Iraqi Islamic Party, a multi-ethnic group seen as moderate Islamist and opposed to violence, said on Monday that there was still room for negotiation on the constitution.
"We have not signed the constitution and we still have the time starting from now until the referendum comes.
"[But]we might say yes to the constitution if the disputed points are resolved," party general-secretary Tariq al-Hashmi told a news conference.
The text read to parliament failed to overcome objections by Sunnis, who lost their political dominance with the fall of Saddam Hussein, despite US efforts to broker a compromise between Iraq's divided ethnic and religious groups.
Earlier, hundreds of people marched in the city of Tikrit to protest against the constitution, witnesses said.
They carried photographs of the former president Hussein and held up banners saying "No to the Zionist-American-Iranian constitution". Some Iraqis say the Shia-led government is too close to Iran.
Other banners proclaimed rejection of "federalism, sectarianism and racism", while the crowd also chanted support for Shia cleric Muqtada al-Sadr, who too opposes a federal structure for Iraq.
"We might say yes to the constitution if the disputed points are resolved."
Iraqi Islamic Party general-secretary
Sunni negotiators have rejected several articles in the constitution which they say threaten the unity of the country, notably those concerning federalism.
Iraqi police did not intervene to break up the demonstration, which was monitored by US helicopters hovering overhead for about two hours.
On Friday, hundreds of former government supporters demonstrated against the constitution in Baquba and Kirkuk, north of Baghdad, while Iraqi leaders were still negotiating over the text.
In Falluja, Sunni Arab tribal chiefs expressed concern that "the constitution is weak and it casts the Sunnis aside" to American officials at the US office in the city.
"We have to keep Iraq united from [Kurdish] Zakho to [Shia] Basra," said one chief.
Sunni tribal chiefs fear the
charter will split the country
"We are ready to participate in the referendum on the constitution - this time, even if there is intimidation, we will still vote," said Shaikh Majid Jasim al-Shuwah, head of the Jumela tribe.
The Sunnis, who largely boycotted January national elections in Iraq, are upset with provisions in the draft charter on federalism and the powers of the president that were finalised on Sunday.
Due to weak representation in parliament, the power of the Sunnis to impact the process has been limited. But now they have the chance to defeat the charter at a scheduled mid-October referendum.
While such an outcome would be a severe blow to the US, the opposition to the charter appears to have as its side-effect one of the US-led forces' key goals - increasing Sunni political engagement.
According to Shaikh Majid and other tribal chiefs at the meeting, Falluja's Sunnis were turning out in force to register themselves on the voting rosters.
"Everybody is registering to vote in the referendum - this constitution is against Iraq as a united Arab Islamic nation," Shaikh Majid said as another tribal counterpart told of how Falluja's "imams are encouraging everybody at prayers every Friday to vote".
Falluja's Sunnis are turning out
in force to register to vote
In the "city of the mosques", a symbol of the Sunni Arab uprising against US-led forces before a massive military assault to root out fighters in November, voter participation encouraged by community leaders such as imams and tribal chiefs marks a massive shift.
US officials say hundreds have been going to sign up each day since voter registration centres opened two weeks ago in Falluja - a claim backed up by residents.
"We're very worried about our future situation - all the Sunnis are going to vote because last time we lost our power in Iraq," said Ahmad al-Ali, a 44-year-old doctor who did not vote in national elections but has registered his name for the referendum.
Ali said that the charter writers were agents of neighbouring countries, in particular Iran, and that Sunnis were registering despite posters going up in the town threatening participants with death.