A member of the leader's parliamentary bloc, however, said Friday that if al-Abadi gets a majority in the legislature he will form the new Cabinet.
Meanwhile, Abadi expressed his respect for "the directives of the Supreme Religious Marjiyah", the Shiite religious authority headed by the spiritual Shiite cleric Ayatollah Ali al-Sistani.
After meeting last Monday for the first time since May's election, parliament has put off its next meeting until September 15, having failed on Monday to elect a speaker, which ought to have been the first step in forming a new government.
"I renew the call for the political blocs in the parliament to accelerate the nomination of the speaker of the house, then the presidency of the republic", Abadi said. The legislator spoke on condition of anonymity because he was not authorized to brief media.
Iraqi Prime Minister Haider Al Abadi looks increasingly unlikely to secure a second term as former allies and leading figures blame him for the fallout from violent protests in Basra. "I did not and will not request the post of prime minister in the second term", he said at a news conference in Baghdad on Thursday.
Oil-rich Basra and other cities in Iraq's southern Shia heartland have been protesting since July over endemic corruption, soaring joblessness and poor public services.
Political analyst Ihsan al-Ashaari said al-Abadi's statement does not mean that the outgoing prime minister is no longer trying to hang on to the post.
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.
"The negotiations are still ongoing", al-Ashaari said.
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