With most votes counted, a bloc headed by cleric Moqtada al-Sadr and another led by a militia leader are ahead, voting officials are quoted as saying.
The ballots of some 700,000 security personnel who voted and Iraqis overseas were yet to be tallied up, meaning Abadi could get a boost five months after he announced victory over Daesh, in a voting that saw a turnout of 44 percent.
Sadr and Amiri both came in first in four of the ten provinces where votes were counted, but the cleric's bloc won significantly more votes in the capital, Baghdad, which has the highest number of seats. It says it will announce the remaining results Tuesday.
Amiri initially formed an alliance with Abadi, but later they announced that they would run on separate ballots, with the possibility of forming an alliance after the elections, in order to form a government. It is also possible for al-Sadr and al-Abadi to join forces which could result in al-Abadi being named prime minister again. Then came the fallout over the failed independence movement, which resulted in great a friction with Baghdad.
Abadi's supporters and opponents had considered his ticket the most likely to prevail in the election and secure him a second term as prime minister.
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The Islamic Republic of Iran welcomes the successful holding of Iraq's parliamentary election in a secure and calm atmosphere and considers it as the continuation of Iraq's remarkable victories against terrorism, the statement said.
Widespread disillusionment with Iraq's current political class appears to have helped influential Shiite cleric Muqtada al-Sadr become the early front-runner in national elections marked by record low turnout.
The 44-year-old derives much of authority from his family, inheriting millions of followers from his father, a revered grand ayatollah killed by the regime of former strongman Saddam Hussein. "As we work together, we follow the principle of partnership to ensure safety and harmony between allies", Iraqi Ministry of Defense spokesman Tahseen al-Khafaji said late last month.
The nationalist cleric's success in the election dealt a blow to Iran, which has steadily increased its influence in Iraq since a USA -led invasion toppled Saddam Hussein in 2003.
Next in the running was the Conquest Alliance, made up of ex-fighters from mainly Iran-backed paramilitary units that battled Islamic State, with results putting them ahead in four provinces and second in eight others. His strong showing tapped into a deep vein of anger among Iraqis over poverty, ineffective public services, lack of jobs and rampant corruption. He had no powerful political machine of his own when he took office.