Their effort received a massive boost from President Barack Obama yesterday.
The US leader stopped short of calling for Nouri al-Maliki to resign, saying "it's not our job to choose Iraq's leaders." But, his carefully worded comments did all but that.
"Only leaders that can govern with an inclusive agenda are going to be able to truly bring the Iraqi people together and help them through this crisis," Obama declared at the White House.
The Iraqi leader's moves last year to crush protests by Sunnis complaining of discrimination under his Shiite-led government sparked a new wave of violence by militants, who took over the city of Fallujah in the western, Sunni-dominated province of Anbar and parts of the provincial capital Ramadi.
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Shiite politicians familiar with the secretive efforts to remove al-Maliki said two names mentioned as possible replacements are former vice president Adel Abdul-Mahdi, a French-educated economist who is also a Shiite, and Ayad Allawi, a secular Shiite who served as Iraq's first prime minister after Saddam's ouster.
Another Shiite from the Supreme Council who is trying to land the job is Bayan Jabr, a former finance and interior minister under al-Maliki's tenure, according to the politicians, who spoke on condition of anonymity.
An Iraqi Shiite lawmaker, Hakim al-Zamili, said he was aware of a meeting in recent days between Iraqi political leaders and US officials over the issue of al-Maliki's future, though he did not know who attended the meeting.
"My view is that safeguarding Iraq is now our top priority," al-Zamili said. "We will settle the accounts later.