The Shia bloc that took the most votes in last month's Iraqi election says it has begun moves to form a government of national unity.
The insurgency draws support from Sunni areas
The United Iraqi Alliance has set up committees to hold talks with Kurdish and Sunni groups in the new parliament.
It says it will form a coalition with Sunni factions but only if they do more to combat the insurgency.
The United Iraqi Alliance failed to win an absolute majority in December's vote and so must govern in a coalition.
The alliance took 128 of the 275 seats, Kurdish parties 53 and the main Sunni Arab bloc 44 in the 15 December poll.
Some Sunnis still allege poll fraud and may challenge the result.
The BBC's Jonny Dymond in Baghdad says there has been a flurry of positive feeling about the idea of a national unity government.
United Iraqi Alliance 10 seats short of a majority
Kurdistan Alliance again likely coalition partner
Sunni Arabs gain much greater representation
Secular alliances win fewer seats
Former PM Iyad Allawi's bloc loses half its seats
Deputy PM Ahmed Chalabi's alliance wins no seats
Hussein al-Shahristani, deputy speaker of the current parliament and a senior member of the UIA, said the party would be happy to review the Iraqi constitution, a key demand of the Sunnis.
But he warned the Sunni parties that if they wanted to join the coalition, they would have to fight the insurgency actively.
"We'll require them not only to condemn terrorism - as they do normally - but to work with us in combating terrorism and overcoming it," he said.
The frontline of the battle against insurgents was often in Sunni areas, he added.
Two leading Sunni politicians on Saturday expressed their interest in joining a coalition government.
"We will participate actively in the political process and we will cooperate with many political entities that share our principles," said Adnan al-Dulaimi, whose Iraqi Accordance Front won 44 seats.
Saleh al-Mutlaq, leader of a separate Sunni group that won 11 seats, made a similar pledge.
However both called for an end to policies by the outgoing Kurdish-Shia coalition which they say victimise Sunni Muslims.
Many in Iraq believe that Sunni involvement in the political process could help bring an end to the violence which has racked the country since 2003, our correspondent says.
IRAQ ELECTION FACTS
275-seat Council of Representatives will have four-year term
18 provinces are taken as separate constituencies
230 seats allocated according to population
45 seats distributed to parties whose ethnic, religious or political support is spread over more than one province
15 million eligible voters
One third of candidates in each party must be women
In other developments:
- Also on Saturday, the Kurdish regional parliament in northern Iraq approves plans to unify the administrations of the two main Kurdish groups within two years
- A car bomb explodes in a crowded Baghdad market, killing one person and wounding half a dozen, police say
- Five members of President Jalal Talabani's staff are injured when a roadside bomb is set off in northern Iraq, although the president is not with the party
- Three policemen are killed in a car bomb attack in Baquba, 55km (35 miles) north-east of Baghdad.