Grand Ayatollah Ali Sistani, who commands unswerving loyalty from many Shi'ites in Iraq and beyond, said political blocs should agree on the next premier, parliament speaker and president before a newly-elected legislature meets on Tuesday.
Sistani's intervention makes it difficult for Maliki to stay on as caretaker leader as he has since a parliamentary election in April. That means he must either build a coalition to confirm himself in power for a third term or step aside.
Sistani's message was delivered after a meeting of Shi'ite factions including Maliki's State of Law coalition failed to agree a consensus candidate for prime minister.
The United States and other countries are pushing for a new, inclusive government to be formed as quickly as possible to counter the insurgency led by an offshoot of al Qaeda, the Islamic State in Iraq and the Levant.
The embattled Maliki accused his political foes of trying to prevent parliament from meeting on time and stirring up violence to interfere with the political process.
"They worked to postpone the elections... and now they are working to postpone the first session of the council of representatives... but if they are not able to pressure us to postpone, they will go for inciting security incidents in Baghdad," he said during a televised meeting with commanders.
Over the past fortnight, militants have overrun most majority Sunni areas in north and western Iraq with little resistance, advancing to within an hour's drive of Baghdad.
Iraq's million-strong army, trained and equipped by the United States at a cost of some $25 billion, largely evaporated in the north after the militants launched their assault with the capture of the city Mosul on June 10.
Thousands of Shi'ite volunteers have responded to an earlier call by Sistani for all Iraqis to rally behind the military to defeat the insurgents.
Under Iraq's governing system put in place after the fall of Saddam Hussein, the prime minister has always been a Shi'ite, the largely ceremonial president a Kurd and the speaker of parliament a Sunni. Negotiations over the positions have often been drawn out: after the last election in 2010 it took nearly 10 months for Maliki to build a coalition to stay in office.
Divvying up the three posts in the four days before parliament meets, as sought by Sistani, would require leaders from each of Iraq's three main ethnic and sectarian groups to commit to the political process and swiftly resolve their most pressing political problems, above all the fate of Maliki.
"What is required of the political blocs is to agree on the three (posts) within the remaining days to this date," Sistani's representative said in a sermon on Friday, referring to Tuesday's constitutional deadline for parliament to meet.
Maliki, whose Shi'ite-led State of Law coalition won the most seats in the April election, was positioning himself for a third term before the ISIL onslaught began. His closest allies say he still aims to stay, but senior State of Law figures have said he could be replaced with a less polarising figure.
Sunnis accuse Maliki of excluding them from power and repressing their sect, driving armed tribal groups to back the insurgency led by ISIL. The president of Iraq's Kurdistan region has also said Maliki should go.
A western diplomat, speaking on condition of anonymity, predicted that Maliki was now done.
"It looks like the debate is whether it is going to be Tareq Najem from inside State of law or someone from outside Maliki's alliance," the diplomat said, referring to Maliki's one-time chief of staff and a senior member of his Dawa party.
"It is generally understood it will not be Maliki," the diplomat said. "Security was his big thing, and he failed."
Allies of Maliki said Sistani's call for a quick decision was not aimed at sidelining the premier, but at putting pressure on all political parties not to draw out the process with infighting as the country risks disintegration.
The Kurds have yet to agree a candidate for president and the Sunnis are divided among themselves over the speaker's post.
Iraqi helicopters fired on a university campus on Friday in Tikrit, the hometown of former dictator Saddam Hussein and second major city to fall to insurgents more than a fortnight ago. Government commandos launched an airborne assault on the campus on Thursday, a rare push back into rebel-held territory.
"My family and I left early this morning. We could hear gunfire, and helicopters are striking the area," said Farhan Ibrahim Tamimi, a professor at the university who fled Tikrit for a nearby town.
ISIL fighters' dramatic advance after capturing the main northern city Mosul on June 10 has placed Iraq's very survival as a state in jeopardy, threatening to reignite the wholesale sectarian slaughter that saw at least 100,000 Iraqis killed during U.S. occupation from 2003-2011.
U.S. President Barack Obama has ruled out sending ground troops back, but has sent up to 300 advisers, mostly special forces, to help the government fight the insurgents.
ISIL fighters who aim to set up a caliphate on both sides of the Syria-Iraq border consider all Shi'ites heretics deserving death. They proudly boasted of executing scores of Shi'ite government soldiers captured in Tikrit.
New York-based rights group Human Rights Watch said on Friday analysis of photographs and satellite imagery indicated ISIL had killed as many as 190 men in at least two locations over three days after they captured Tikrit. The death toll may be much higher but the difficulty of locating bodies and getting to the area had prevented a full investigation, it added.
However, there have also been accounts of government forces killing large numbers of prisoners. Several police officials told Reuters 69 prisoners had been killed on Monday while being transported from a jail in Hilla south of Baghdad. Last week 52 prisoners were killed in a jail in Baquba to the north.
In both cases the official account was that prisoners died in custody in the crossfire during insurgent attacks.
Amnesty International also said it had gathered evidence pointing to a pattern of extrajudicial executions of detainees carried out by government forces before withdrawing from cities, including Tal Afar, west of Mosul, which militants now control.
Fighters from ISIL have been joined by other, less radical groups who share their view that Sunnis have been persecuted under Maliki. The onslaught has been halted outside the capital, but militants have continued to advance and consolidate their gains elsewhere, including the area around Mosul in northwestern Iraq, which is home to many religious and ethnic minorities.
Militants took control of six villages populated by the country's Shi'ite Shabak minority southeast of Mosul after clashing with Kurdish "peshmerga" forces securing the area, according to a lawmaker and community leader.
Up to 10,000 people have also fled from the predominantly Christian communities of Qaraqosh, some 30 kilometres southeast of Mosul, since Wednesday, fearful of becoming a target for ISIL, the U.N. refugee agency said.