The most serious threat to Mr Abadi is Hadi al Ameri, a former commander of Iran-backed units that fought IS.
Iraqi Prime Minister Haider al-Abadi (C) speaks as he presents candidates running on his list during an electoral campaign rally in the holy Iraqi city of Karbala on May 4, 2018.
Saturday's elections are the first in Iraq since the defeat of Daesh a year ago by Iraqi forces backed by a US-led coalition.
The official turnout was expected to be low as numerous roughly 24.5 million voters skipped the parliamentary ballot, despite it passing off largely peacefully.
The ballot Saturday saw a record low turnout, as only 44.5 percent of eligible voters headed to the polls in the lowest participation rate since the 2003 US-led ouster of Saddam Hussein. Full results are due to be officially announced on Monday.
Al-Abadi sought to retain his post as prime minister after overseeing the military defeat of the IS movement, but faced stiff competition from his predecessor, Nouri al-Maliki, as well as al-Sadr and the Fatah alliance of candidates with paramilitary ties.
Entrenched corruption, the influence of Iran and the future of USA forces now in Iraq are other issues that have dominated the run-up to the election.
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After polls closed, Abadi said in a statement that the Iraqis had been able to vote "freely and safely". Sadr's father, highly respected Grand Ayatollah Mohammed Sadeq Al Sadr, was murdered in 1999 for defying Saddam Hussein.
According to the officials' latest announcement, the political coalition of Shia Muslim cleric Muqtada al-Sadr has an early lead.
Seats in parliament will be allocated proportionately to coalitions once all votes are counted.
Reuters could not independently verify the document's authenticity but the numbers in it, which showed the results from all 18 provinces, matched those of the electoral commission in the 10 provinces for which it has announced results. Al-Sadr commanded fighters in the war against the IS group and headed a powerful militia that fought USA forces in Iraq prior to 2011, but his 2018 campaign focused on social issues and eliminating government corruption.
The new president will within 15 days formally task the largest parliamentary bloc's nominee with forming a government.
The US-led coalition that helped battle IS pledged Sunday to work with the elected government to ensure the "lasting defeat" of IS and said the poll proved Iraqis "emphatically rejected violent extremism". Dozens of alliances ran for office in these elections and months of negotiations are expected before any one alliance can pull together the 165 required seats.