The Saeroon Alliance led by al-Sadr, the Shia who many U.S. military officials hold responsible for the deaths of USA troops, won by taking the most number of seats.
It comes amid mounting fraud allegations surrounding the 12 May elections, when Sadr was announced the surprise victor of the vote.
Meanwhile, Iraq's Independent Higher Election Commission has announced that all the ballot boxes in the warehouse are safe.
A fire truck arrives at the scene of a blaze Sunday at a warehouse in Baghdad storing ballots from Iraq's parliamentary elections.
The political system is created to ensure that no one person or party can dominate, which leads to post-election discussions, a move that has not been interrupted by parliament's decision to recount.
According to intelligence services, tests of electronic voting machines - used for the first time in Iraqi elections - produced varied results, appearing to give credence to the fraud claims.
It won 54 of the 328 seats in parliament - making it the largest bloc in the post election landscape.
Abadi, whose electoral alliance came third in the election, had said on Tuesday that a government investigation had found serious violations and blamed Iraq's independent elections commission for majority.
Sunday's fire broke out at a Trade Ministry site in Baghdad where the election commission stored the ballot boxes from Al-Rusafa, the half of Baghdad on the eastern side of the Tigris river.
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The previous week parliament had already voted to annul the ballots of displaced Iraqis and those living overseas, although they accounted for only a small fraction of the overall vote.
Top Sadr aide Dhiaa al-Asadi said the fire was a plot aimed at forcing a repeat of the election and hiding fraud.
Jabouri's call was seconded by Vice President Iyad Allawi, the leader of the electoral alliance Jabouri ran as part of.
But he has more recently positioned himself as an Iraqi nationalist, critical of both Iranian and American influence and sectarian conflicts in Iraq, and forming diverse alliances with communists, Sunnis and independents. Only one - housing electronic equipment and documents - had burned down, he said.
"It is possible there were also some ballot boxes in the warehouse that caught fire, but most of the important boxes are in the three warehouses, where the fire has been controlled", he said.
The ballot boxes are needed for a manual recount of votes, mandated in a law passed by the Iraqi parliament on Wednesday.
Interior ministry spokesman General Saad Maan echoed him during a visit to the scene to the warehouse where the blaze was raging.
An official statement said a recent cabinet meeting chaired by the premier had named the Iraqi anti-graft chief as the head of the commission.