The Sunnis had walked out of the talks last week after one of their committee members was shot dead in front of a restaurant.
At crisis talks on Monday they secured pledges of better security and an investigation into the assassination.
“We will definitely return tomorrow,” said Saleh Mutlaq, spokesman for the Sunni umbrella group Iraqi National Dialogue, which slain committee member Mujbil al-Sheikh Isa belonged to.
Abdul Nasser al-Jenabi, a committee member from another Sunni group, said its demands had been met. The government announced the compromise in a statement signed by parliament Speaker Hajem al-Hassani.
Khalilzad says the constitution
Iraq’s government and its US sponsors had hoped the presence of Sunnis on the constitution-drafting team would help to defuse violence.
Fifteen Sunni members were drafted on to the committee last month, joining members drawn from a parliament mainly made up of Shia and Kurds – elected in a January vote when most Sunnis stayed at home because of a boycott or fear of reprisals.
US Ambassador Zalmay Khalilzad said a constitution was “important in terms of weakening the insurgency [and] winning the population away from al-Qaida and the other foreign terrorists who come here to kill and maim and use Iraqis as cannon fodder for their larger agenda”.
The bloodshed has made political agreement more urgent.
Two vehicle bombs struck police checkpoints in Baghdad on Monday morning, killing at least 15 people, while funerals were held for some of the victims of a huge truck bomb on Sunday.
Two vehicles bombs killed 15
The US military said 40 people were killed in that attack, although police did not confirm the figure.
The end of the Sunni boycott will allow politicians to breathe a sigh of relief, but negotiations over the constitution are far from over. A draft is due by 15 August, but 1 August is the deadline for announcing a 6-month extension if the committee decides it needs more time.
Kurdish committee member Mahmoud Othman said that, even if the Sunnis rejoined, “I very much doubt at this stage that we are going to have a document ready by the end of this month”.
The sides are divided on issues of federalism – how to share power and resources in areas such as the mainly Kurdish north and the Shia south, where local leaders want autonomy from Baghdad and control of oil wealth.
Other stumbling blocks include the role of religion, a decision on Iraq’s official languages and women’s rights.
“I very much doubt at this stage that we are going to have a document ready by the end of this month”
Washington says an extension would deprive the political process of momentum.
But Othman said: “Nobody likes to talk about requesting an extension, because no one wants to be the one who is responsible for asking for it. But everyone is thinking about it.”
Khalilzad said he did not expect a delay.
Attacks, especially car bombings, rose after the government took power in April, and after a brief lull have intensified again in the past two weeks, killing more than 200 people in Baghdad and towns to the south.
At dawn on Monday, a bomber blew up a minivan packed with explosives at a checkpoint near the Sadir hotel in the city centre. A second bomber struck Ansour Square, near an entrance to the fortified Green Zone government and diplomatic compound.
The Defence Ministry said 12 people died in the first strike. Hospital sources said at least three died in the second.
Those bombs followed Sunday’s huge blast, the worst in more than a week, when a suicide truck bomb packed with more than 226kg of explosives blew up near a police station.
The US Army, citing Iraqi police estimates, put the toll at 40. But police sources told Reuters that 25 were killed.
US forces said one soldier was killed by a roadside bomb on Monday in Samarra, a Sunni stronghold north of Baghdad. In Baghdad’s Dora neighbourhood, four people were killed, including two women, when armed men stormed their house, police said.