The sources cited unofficial initial results.
Iraqis voted on Saturday in the first election since the defeat of Islamic State militants inside the country. Final results are expected on Monday.
Turnout was 44.52 percent with 92 percent of votes counted, the Independent High Electoral Commission said - significantly lower than in previous elections. Results are due to be officially announced on Monday.
Abadi, a rare ally of both the US and Iran, was mainly concerned with fending off Shi'ite Muslim groups other than Sadr's alliance, which are seeking to pull the country closer to Tehran.
Those rivals were his predecessor as prime minister, Nuri al-Maliki, and the leader of the main Shi'ite paramilitary group, Hadi al-Amiri, both closer than he is to Iran, which has wide sway in Iraq as the primary Shi'ite power in the region.
Unofficial results compiled by Reuters reporters in southern provinces also indicated that Sadr, a firebrand cleric who led a violent uprising against US troops from 2003-11, appeared to be making a strong showing.
If the Sadr list finished second, that would mark a surprise comeback by the cleric. Sadr has a zealous following among the young, poor and dispossessed but has been sidelined by influential Iranian-backed figures such as Amiri. Sadr has kept Tehran at a distance.
Whoever wins the election will have to contend with the fallout from US President Donald Trump's decision to quit Iran's nuclear deal, a move Iraqis fear could turn their country into a theater of conflict between Washington and Tehran.