Prime Minister Haidar al-Abadi came in first in Iraq's second largest province, Nineveh, but he has performed poorly in the rest of the country, coming in third and fourth place in most provinces and fifth in Baghdad.
Iraq's electoral commission says a coalition led by Shia leader Muqtada al Sadr has won the election, with 91 percent of the ballots counted.
The alliance is headed by Hadi al-Amiri, a former minister of transport with close ties to Iran who became a senior commander of paramilitary fighters in the fight against the Islamic State extremist group.
Sadr - the former leader of the anti-Western Mahdi Army during the USA occupation - ran on a campaign promising to stamp out corruption but has also been keen to distance himself from Iranian influence. The country is also waiting to hear results from Kirkuk, an oil-rich city disputed by Baghdad and Iraq's northern, autonomous Kurdish administration.
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Al-Sadr led the Mahdi army in the early years of the war on terror, using IEDs and machine guns to kill Western forces, O'Neill said.
The younger al-Sadr campaigned on a cross-sectarian platform of fighting corruption and investing in services and struck a surprising alliance with the Communist Party in the capital.
On Tuesday, the prime minister called Sadr to congratulate him for the election victory, the cleric's office said.
Mr Al Sadr made a concerted effort to appeal to marginalised and disillusioned Iraqis while styling himself as a reformist leader determined to effect comprehensive change to a failing political class. Whoever wins the most seats must negotiate a coalition government in order to have a majority in parliament.
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Any attempt to form a government that would threaten the influence Iran has built up in the 15 years since the fall of Saddam Hussein looks certain to face opposition from Tehran.
The final results are due to be announced later on Monday, triggering what are expected to be lengthy negotiations to form a new coalition government.
According to Agence France Presse, Iran is holding meetings to try to block Shiite cleric Moqtada Sadr, who heads Iraq's Sairoon coalition, from forming a government.
The remaining uncounted ballots, mostly from Iraqis overseas, the security services, and internally displaced people voting in camps and elsewhere, might change the final seat tallies but only marginally. Iran has publicly stated it will not allow his bloc to govern.
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