Security and commission sources had earlier said Abadi was leading the election, which was held on Saturday and is the first since the defeat of Daesh group in the country.
Turnout was 44.52%, the Independent High Electoral Commission said, significantly lower than in previous elections. The commission gave no indication on when further results would be announced. Incumbent Prime Minister Haidar al-Abadi, who had sought to reach across Iraq's sectarian divides, was trailing.
Election officials said that full final results could be announced in the next 24 hours.
A document provided to Reuters by a candidate in Baghdad that was also circulating among journalists and analysts showed results from all 18 provinces.
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If the Sadr list finished second, that would mark a surprise comeback by the cleric.
Mr Sadr came to prominence after the fall of Saddam Hussein in 2003, as the leader of a group of young men in a largely poor area of Baghdad.
The strong showing could be a testament to al Sadr's loyal base of followers he maintains who cast their ballots despite a general mood of apathy that kept many Iraqis away from the polls. Sadr has recently campaigned against corruption and can summon millions into the streets to protest policies he opposes.
He derives much of his authority from his family.
Celebrations erupted in Baghdad's Sadr City, an impoverished quarter that is home to some three million people and is named after the cleric's late father, Ayatollah Mohammad Sadq al Sadr.
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Second place was taken by the force of Shiite militia leader Hadi al-Amiri, who Iran supports.
Whoever wins the election will have to contend with the fallout from U.S. president Donald Trump's decision to quit Iran's nuclear deal, a move Iraqis fear could turn their country into a theatre of conflict between Washington and Tehran.
If parliament chooses him as prime minister, Abadi will remain under pressure to maintain that balancing act amid tensions between Washington and Tehran over the nuclear accord.
The election was the first since the government declared victory over so-called Islamic State (IS) a year ago.
Among the traditional powerbrokers looking set to lose big at the election was divisive former premier Nuri al-Maliki, who remains widely reviled for the loss of territory to IS.
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