Cleric Muqtada al-Sadr, whose bloc won the 12 May elections, has distanced himself from his one-time ally Abadi, while a rival alliance of pro-Iranian former paramilitary fighters led by Hadi al-Ameri said it would work with Sadr to form a new government that excludes the premier.
Embattled Iraqi Prime Minister said Haider al-Abadi said he is not going to hold on to power.
The outgoing Iraqi prime minister also called on political blocs to speed up their nomination of the three leadership posts of parliament speaker, president and prime minister.
Since July 9, Iraq's Shia-majority central and southern provinces - especially Basra - have been rocked by protests to demand better public services, more job opportunities and an end to government corruption.
"I did not and will not request the post of prime minister in the second term", Anadolu Agency quoted Abadi as he addressed a news conference in Baghdad.
Queensland offers R1m reward as strawberries sabotaged with needles
NSW Police have so far received reports of contaminated strawberries bought from supermarkets in Tweed Heads, Taree and Wingham. One Australian woman posted photos online showing the needles inside strawberries her 10-year-old son was eating.
Al-Abadi, who was backed by the US for a second term in office, said he would respect the directives of Grand Ayatollah Ali al-Sistani, who said it was time for Iraq to appoint a prime minister it has never had before.
Iraq's newly elected parliament held its first session as two blocs, both claiming to hold the most seats, vied for the right to form a new government.
Mr Sadr's election list came in first in the polls, winning 54 seats in the 329-seat legislative body, while Mr Abadi's list came in third, with 42 seats.
Mohammed Zayni, the eldest member of parliament who temporarily chairs the newly elected parliament, made a decision to resume the parliament session on September 15 to elect a new speaker and two of his deputies.