The appeal by Grand Ayatollah Ali al-Sistani comes as al-Maliki is fighting to keep his job, with even key patron Iran exploring alternatives in the face of Iraq's worst crisis since US troops withdrew at the end of 2011.
The conflict has drawn the Americans back to Iraq with special forces being deployed to help Iraqi troops. The US also has started flying armed drones over Baghdad to protect US interests in the Iraqi capital, a Pentagon official said Friday. He spoke on condition of anonymity because he was not authorized to discuss the new flights on the record.
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Al-Maliki, who has governed the country since 2006, needs support from other parties after his State of Law bloc won the most seats in the elections but failed to gain the majority needed to govern alone. That set the stage for potentially months of coalition negotiations. But now a new government is wanted urgently to face the lightning advance across the north and west of the country by the al-Qaida breakaway Islamic State of Iraq and Syria.
The stunning gains were made possible in large part because Iraqi security forces melted away in the face of the onslaught.
Human Rights Watch released a report Friday about the killings of scores of police and soldiers by the Sunni militants in the days after it captured the northern city of Mosul on June 10 then stormed south to capture Saddam Hussein's hometown of Tikrit.
The killings were widely reported after the Islamic State posted graphic, online photos showing dozens of men wearing civilian clothes lined up and bent over as militants pointed rifles at them from behind. A final set of photos shows bodies.
Human Rights Watch said that based on analysis of the photos and satellite imagery, the militants killed between 160 to 190 men in two locations in Tikrit between June 11 and June 14.
"The number of victims may well be much higher, but the difficulty of locating bodies and accessing the area has prevented a full investigation," the group said.
Human Rights Watch said it used satellite imagery from 2013 and publicly available photos taken earlier to pinpoint the site of the killings in a field next to the Tigris River and near one of Saddam's former palaces. It said satellite imagery of the site from June 16 did not reveal bodies but showed indications of earth movement consistent with the two shallow trenches visible in the photos, in which the soldiers were forced to lie down before being shot.
"The photos and satellite images from Tikrit provide strong evidence of a horrible war crime that needs further investigation," Peter Bouckaert, emergencies director at Human Rights Watch, said in a statement.
Chief Iraqi military spokesman Lt. Gen. Qassim al-Moussawi confirmed the online photos' authenticity on June 15, after they first surfaced, and told The Associated Press at the time that an examination of the images by military experts showed that about 170 soldiers were shot to death after their capture.
Captions on the photos showing the soldiers after they were shot say "hundreds have been liquidated," but the total could not be verified.
The massacre appeared to be aimed at instilling fear in Iraq's demoralized armed forces as well as the country's Shiite majority, whom the Islamic State views as apostates.
But evidence points to atrocities committed by both sides in the conflict, rights groups say.
Amnesty International said Friday that it has evidence that suggests "a pattern of extrajudicial executions" of mainly Sunni detainees by government forces and Shiite militias in Tal Afar, Mosul and Baqouba. It based its conclusions on interviews with survivors and relatives of those killed.
The Baqouba killings, in which Shiite militiamen shot dead nearly four dozen Sunni detainees in a city jail, have been widely reported. Amnesty said its report shows "a worrying pattern of reprisal attacks against Sunnis" in retaliation for the insurgent gains.
On the military front, a senior Iraqi army official told the AP that four helicopters carrying Iraqi commandos landed at a soccer pitch inside a university campus in militant-held Tikrit late Thursday and clashed with Islamic State fighters for several hours.
One of the helicopters developed mechanical problems after takeoff from Tikrit, 80 miles (130 kilometers) north of Baghdad, but landed safely in the provincial military headquarters. The official had no word on casualties and declined to specify the mission's objectives. He spoke on condition of anonymity because he was not authorized to brief the media.
The official also said 200 troops have arrived at a key refinery north of Baghdad under attack by militants for more than a week. The reinforcing troops join a 100-strong contingent that has been defending the Beiji refinery, Iraq's largest and the source of about a quarter of the country's oil product needs, including fuel for power stations.
State-run television aired footage Friday purporting to show troops disembarking from helicopters at Beiji, with some carrying boxes of supplies. Dense black smoke was rising from what appeared to be a large fuel tank.
The news from Tikrit and Beiji suggested that Iraq's military was stepping up efforts to regain its footing against the insurgents, who appear to be trying to carve out a self-styled Islamic state astride the Iraqi-Syrian border.
On Friday, al-Maliki warned army commanders that militants were likely to try to undermine security in the Iraqi capital ahead of Tuesday's parliamentary session. "Baghdad must be secured and not subjected to any instability at this time," he said in televised comments.
Al-Maliki sounded upbeat on the military situation, saying the armed forces have regained the initiative and were now on the offensive. He cited Thursday's Tikrit raid as an example. He also vowed to severely punish army commanders whom he said were taking bribes from soldiers to allow them to stay home.
The United States and other world powers have pressed al-Maliki, in office since 2006, to reach out to the country's Sunni and Kurdish minorities and have called for a more inclusive government that can address longstanding grievances. But he has instead widely been accused of monopolizing power and alienating Sunnis, and his failure to promote national reconciliation has been blamed for fueling the Sunni anger.
Al-Sistani also called on Iraq's politicians to agree on the next parliament speaker and president by the time the new legislature meets on Tuesday, a cleric who represents him told worshippers in a Friday sermon in the holy city of Karbala.
Abdul-Mahdi al-Karbalaie told worshippers that would be a "prelude to the political solution that everyone seeks at the present."
The reclusive al-Sistani, the most revered figure among Iraqi Shiites, rarely appears or speaks in public, instead delivering messages through other clerics or, less frequently, issuing edicts.