But the government formation process, which international leaders and Iraq's top Shiite cleric have urged be expedited, was dealt a blow when a parliament session scheduled for Tuesday was postponed.
Washington insisted that uniting Iraq's sectarian factions was the only way to repel the advances by Sunni jihadists.
White House spokesman Josh Earnest said that Islamic State (IS) radicals posed an "existential threat".
"To confront that threat, the country will need to be united," he said.
Iraqi officials and a lawmaker, all speaking on condition of anonymity, said the parliament meeting was rescheduled for August 12 because MPs could not agree on a new speaker.
More than two months after elections in which Maliki's camp won the most seats, though not a majority, parliament has yet to progress on filling the top three positions, which are split between the Shiite Arab, Sunni Arab and Kurdish communities.
A parliament session last week ended in chaos, with MPs trading heckles and threats before some eventually walked out, forcing an adjournment.
The UN's special envoy, Nickolay Mladenov, has warned that further delays risked plunging the country into "Syria-like chaos".
And he called Monday for political blocs to "limit the number of days for final negotiations, and work within constitutionally mandated timelines for the necessary nominations".
- Maliki seeking third term -
Despite telling AFP in a 2011 interview he would not seek a third term, Maliki vowed last week he would not bow to mounting international and domestic pressure to step aside and allow a broader consensus.
Iraqi forces have largely regrouped after the debacle that saw soldiers abandon their positions and, in some cases, even weapons and uniforms as IS-led militants conquered second city Mosul and advanced to within about 80 kilometres (50 miles) of Baghdad.
But while Iraq has received support including equipment, intelligence and advisers from the United States, Russia, Iran and even Shiite militias it once shunned, efforts to battle the militant offensive were dealt a blow when a senior general was killed on Monday.
Staff Major General Najm Abdullah al-Sudani, commander of the army's 6th division, was killed by "hostile shelling", Lieutenant General Qassem Atta told AFP by text message.
Sudani was killed west of Baghdad, near where security forces have been locked in a more than six-month standoff with militants who have seized the city of Fallujah.
Maliki attended Sudani's funeral along with acting defence minister Saadun al-Dulaimi and senior officers, the defence ministry said.
Security forces have for more than a week also tried to retake the Sunni stronghold of Tikrit from a loose alliance of IS fighters, other jihadist groups and former Saddam Hussein loyalists, but have so far failed to do so.
Iraqi forces have been hamstrung by a lack of combat experience and dearth of intelligence in Sunni areas, largely over distrust of the Shiite-led authorities among minority Sunni Arabs, analysts say.
- 'Collateral damage' -
"The army and the police are seen as sectarian... and therefore the Sunni community doesn't provide support or, crucially, intelligence to the security forces," said John Drake of the AKE Group security company.
"If you don't have good intelligence on the ground, your strikes are not precise, they involve collateral damage and casualties... making everything worse."
Many of the 21 people killed in air strikes carried out in Salaheddin and Nineveh provinces on Sunday were said to have been civilians.
While most observers have argued Baghdad is not about to fall, violence there has continued.
In the latest attack, a suicide bomber struck the northern entrance to the Shiite-majority Kadhimiyah area on Monday, killing at least five people and wounding at least 13, officials said.
And while government forces were still looking for a major victory, IS jihadists appeared to be brimming with confidence.
A few days after declaring the establishment of a "caliphate", the group's leader Abu Bakr al-Baghdadi stepped out of the shadows to deliver a Friday sermon in Mosul's largest mosque.
Analysts have described the sudden public appearance by the self-proclaimed "caliph" -- second only to Al-Qaeda chief Ayman al-Zawahiri on the US most wanted list -- as a daring stunt reinforcing Baghdadi's status as the new strongman in the world of global jihad.