High-profile cleric Moqtada al-Sadr's Moving Forward bloc has a substantial lead.
The early results reported late on Sunday suggested that Al-Sadr's bloc was ahead.
The final results were due to be announced later on Monday.
The Shia-led administration of Mr Abadi has won praise because of the struggle against IS militants, also security has vastly improved throughout the country.
UFC 224 Play-By-Play: Mackenzie Dern Quickly Submits Amanda Bobby Cooper
Dern wanted to make the quick turn around for this fight at UFC 224 because it was in her adopted hometown of Rio de Janerio. Despite the fact that Mackenzie Dern missed weight by seven pounds, her fight against Amanda Cooper at UFC 224 will proceed.
Both Sadr's bloc and the bloc loyal to Hadi al-Amiri, the head of the Badr Brigade, are the top two right now, with Abadi's state of law in third.
On Saturday, Iraqis voted in the country's first parliamentary election since Iraq declared victory against the Daesh terrorist group.
Unlike previous elections, new electronic devices were used in Saturday's parliamentary elections to check election cards, fingerprints, and count ballots-a system meant to streamline the process and prevent fraud from taking place. It included full returns from only 10 of the country's 19 provinces, including the provinces of Baghdad and Basra. Officials said turnout was only 44 percent, the lowest ever since Saddam's ouster.
Iraqi Prime Minister Haider al-Abadi's Victory Alliance is in third place.
Many Iraqis fear that their country can once again turn into a casualty in virtually any conflict involving America and Iran, our correspondent adds.
Paul Ryan Tries To Quietly Put Down House GOP Immigration Rebellion
The bipartisan legislation in the House focuses squarely on DACA and border security other than wall construction. They say the petition would wrest control over the House agenda away from Ryan.
His father, highly respected Grand Ayatollah Mohammad Sadeq al-Sadr, was murdered in 1999 for defying Saddam Hussein.
Al-Sadr's supporters jubilant: After the announcement that Muqtada al-Sadr's movement Marching Towards Reform was leading the polls in Baghdad, supporters took to the streets in the capital to celebrate a win. His father's cousin, Mohammed Baqir, was killed by Saddam in 1980.
Abadi, who came to power four years ago after Islamic State seized a third of Iraqi territory, received USA military support for Iraq's army to defeat the Sunni Muslim militant group even as he gave free rein to Iran to back Shia militias fighting on the same side. It's reportedly supported by Iran.
Abadi was seen by some Iraqis as lacking charisma and ineffective. Al-Sadr is a staunch foe of Iranian and American influence in Iraqi politics.
Political power in Iraq is traditionally divided along sectarian lines among the offices of prime minister, president and parliament speaker.
19 dead in fighting between army and rebels — Myanmar military
He called the operation "terrorism". "They ran away after attacking the civilian targets", said Zaw Htay. But it's hard to say how they were killed during the fighting", said the TNLAA spokesman.