The Election Commission said Al-Abadi's governing alliance is running third.
The Saturday election was the first since the government declared victory over so-called Islamic State (IS) terror group in 2017.
The final results were due to be announced later on Monday.
The complex electoral system in Iraq's 18 provinces suggested that the final make-up of the 329-member parliament was still far from decided, the BBC said.
Turnout in the election was 44.5 per cent -- much lower than in previous elections.
Iraqis voted for rival lists of candidates. Most of whom are predominantly Shia or Sunni, though the Kurds have their own lists.
The early results reported late on Sunday suggested that Al-Sadr's bloc was ahead. His supporters celebrated in Baghdad.
According to the reports, the group led by veteran militia leader Hadi al-Amiri, was in second place.
But despite improved security, Iraq was still struggling to rebuild itself after four years of war against IS, the report said.
Many Iraqis were disillusioned by widespread government corruption and a weak economy, the BBC added.
Some Iraqis fear their country could once again become a casualty in any struggle between the US and neighbouring Iran, the BBC said.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)