Iraqis voted on Saturday in the first election since the defeat of Islamic State militants inside the country.
The Marching Towards Reform alliance of Shiite cleric Moqtada Sadr and communists was ahead in six of Iraq's 18 provinces and second in four others. The group's remnants, though, keep staging sporadic attacks across Iraq.
Amiri leads the Badr Organization, which was the backbone of the volunteer forces that helped to defeat IS along with Iraqi government troops and US -backed Western coalition forces.
He called on Iraq's election commission to "take quick measure to check the boxes and the objected electronic devices and to announce the results to the public to maintain the integrity of the elections". By the end of the announcement, al-Sadr's list had the highest popular vote, followed by al-Amiri's.
Prime Minister Haidar al-Abadi on Sunday ordered Iraqi forces to enforce security in northern Iraq's Kurdish region, following violent protests in the wake of the country's parliamentary polls.
Both Sadr's bloc and the bloc loyal to Hadi al-Amiri, the head of the Badr Brigade, are the top two right now, with Abadi's state of law in third.
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Fatah's strong result will be seen as a victory for Iran as it seeks to protect its interests in the Iraq, including the militias it finances and has sometimes directed to fight alongside its forces in Syria.
The Arab and Turkmen political parties in Kirkuk have given an independent commission in charge of the election 24 hours to start a manual recount, local politician Ershad Salihi, the head of the Iraqi Turkmen Front, said.
Sadr portrays himself as an Iraqi nationalist and past year met Crown Prince Mohammed bin Salman of Saudi Arabia who is staunchly opposed to Iran.
Any political party or alliance must gain a majority of Iraq's 329 seats in parliament to be able to choose a prime minister and form a government.
Until a new prime minister is chosen, al-Abadi will remain in office, retaining all his power. Negotiations on forming a coalition are expected. No election since 2003 has had turnout below 60 percent.