Iraqi Prime Minister Haider al-Abadi said that a manual recount of Saturday's national election results should be held if there is evidence that the new electronic voting system has proven unreliable.
Influential cleric Moqtada al-Sadr's bloc won the most seats, while Badr Brigade head Hadi al-Amiri came in a close second.
After being sidelined by Iranian-backed rivals for years, the apparent parliamentary victory marks a political comeback for Sadr, who didn't even officially run for prime minister in this year's elections.
Sadr - the former leader of the anti-Western Mahdi Army during the U.S. occupation - ran on a campaign promising to stamp out corruption but has also been keen to distance himself from Iranian influence.
The surprising upset in elections this weekend by Mr. Sadr's unlikely alliance of communists, Sunni businessmen and pious community activists comes amid long-simmering anger at the established politicians who have controlled government since Iraq's first democratic election in 2005 after the fall of dictator Saddam Hussein.
Abadi offered a statement of congratulations to Sadr for his victory, and for encouraging a secure atmosphere during the vote.
The initiative could trump Sadr's own coalition-building efforts but it risks angering the cleric's supporters who are yearning for a clampdown on corruption among establishment figures.
None of the competing blocs appears on track to win a majority in parliament and name a prime minister.
The US has called for an "inclusive government, responsive to the needs of all Iraqis".
Seats in parliament will be allocated proportionately to coalitions once all votes are counted. Full results are due to be officially announced later on Monday.
But after 14 years, more than $1 trillion, millions of Iraqi dead and thousands of US troops killed and maimed, the USA now has less influence over Iraq than it had while former Central Intelligence Agency operative Saddam Hussein was in power.
The elections commission has announced results from 16 of Iraq's 19 provinces. Also at issue is the influence of Iran on the country: Iranian-backed Shiite militias who played a key role in defeating IS and were allied with the Shiite-led Baghdad government made significant electoral gains. Iran has publicly stated it will not allow his bloc to govern.