Sadr has led two uprisings against United States forces in Iraq and is one of the few Shia leaders to distance himself from Iran.
Partial returns of the 2018 vote the first since Iraq declared victory over the Islamic State group were announced late Sunday by the Iraqi electoral commission and put al-Sadr's political alliance in the lead in four provinces, including Baghdad.
While incumbent Prime Minister Haider al-Abadi was considered the most likely victor, the alliance of populist Shiite cleric Moqtada al-Sadr, who has very hard relationships with the United States, is the likely victor in a surprise twist, according to preliminary results based on more than 91 percent of the votes cast in 16 of Iraq's 18 provinces.
He can not become prime minister as he did not run in the election, though his apparent victory puts him in a position to pick someone for the job.
But he could rule by proxy.
Populist Shia cleric Sayyid Moqtada Sadr on Tuesday eyed a broad coalition after a shock election triumph that has upended Iraqi politics.
However, Sadr's coalition looks unlikely to win enough seats to form a government in its own right, and is likely to begin talks with other candidates to name a new prime minister.
But with his group set to be far from a majority in parliament, wrangling over any potential coalition should take months - and there remain major obstacles ahead that could thwart Sadr's ambitions.
Saturday's election is the first since the defeat of ISIL previous year. "But we have had no results".
Potraying himself as an Iraqi nationalist, Sadr has a zealous following among the young, poor and dispossessed, but he had been sidelined by influential Iranian-backed figures.
In a sign that he is angling to chart a different course, he visited regional Sunni powerhouse Saudi Arabia past year, as Tehran's rival seeks to play a greater role in Iraq.
Sadr is likely to face fierce opposition from established political forces, who may look to coalesce in a bid to stop him taking control. Former Prime Minister Nuri al-Maliki, a close ally of Iran like Amiri, came in fourth with around 25 seats.
While speculation swirls, the next concrete step remains completing the vote count and firming up the final makeup of Iraq's new 329-seat parliament.