They hope to focus on their primary task - writing a new constitution.
The impasse, two months after the country's historic national elections, is rooted in disagreements about the posts that should be granted to Sunni Arabs in an attempt to incorporate members of the minority group that dominated under Saddam Hussein, in the new government.
The Sunni Arab minority was given until Sunday to come up with a candidate to serve as speaker.
"We saw that things were confused ... so we gave [the Sunnis] a last chance," said Husain al-Sadr, a Shia cleric and member of interim Prime Minister Iyad Allawi's coalition.
"We expect the Sunni Arab brothers to nominate their candidate. Otherwise, we will vote on a candidate on Sunday."
Also on Wednesday, more than 35,000 Shia students from across Iraq gathered in the southern city of Hilla and began marching to Karbala to celebrate the al-Arbain religious festival.
The religious holiday marks the end of a 40-day mourning period for one of the grandsons of Islam's Prophet Muhammad, Imam Husain, who was killed in a 7th-century battle.
"We expect the Sunni Arab brothers to nominate their candidate. Otherwise, we will vote on a candidate on Sunday"
Husain al-Sadr, Shia cleric
Underscoring tensions with the country's majority Shia, fighters set off two explosions on Monday that targeted pilgrims.
In Musayyib, 65km south of Baghdad, a bomber on a bicycle blew himself up near a police patrol protecting the pilgrims, killing two policemen.
The other bombing took place at the Imam al-Khadhir shrine compound in Khalis, 80km north of Baghdad. The attack killed one pilgrim and wounded two others.
The Shia-led United Iraqi Alliance party, along with the Kurdish coalition, want a Sunni Arab to take the parliament speaker post as a way to bring more Sunnis into the new government.
Many Sunnis boycotted the 30 January elections or simply stayed home because they feared attacks at the polls.
Officials lobbied prominent Sunni Arab leader and interim President Ghazi al-Yawir, to take the job. But he held out for one of two vice-presidential spots.
The Sunni-nominated candidate, Adnan al-Janabi, was vetoed by the Shia and Kurds due to his brother's ties to Saddam Hussein's former government.
Some politicians speculated that the delay could force them to request a six-month extension to the 15 August deadline for drafting the country's permanent constitution.
But most were opposed to the idea at this stage.
Al-Yawir has rejected the role of
The assembly still needs to name an Iraqi president and two deputies, who will in turn nominate a prime minister.
The presidency is expected to go to Kurdish leader Jalal Talabani and the premiership to Shia politician Ibrahim al-Jafari.
Negotiations over the top posts in the different ministries also continues, with both the Kurds and Shia vying for the Oil Ministry position.
Some Sunnis hope to get the Interior Ministry post, but the Alliance wants them to have the Defence Ministry position instead.
Together, the Alliance and the Kurdish coalition have 215 seats in the 275-seat National Assembly - enough to push through their proposals.
But they have been reluctant to alienate the Sunni Arabs and other minority groups, saying they want an inclusive government.