The deaths threaten to reignite an unprecedented protest movement against government graft and incompetence that erupted across Baghdad and southern Iraq in October but had waned in recent months.

On Sunday, demonstrators staged angry rallies in the capital and several southern cities, where temperatures topping 50 degrees Celsius (122 Fahrenheit) have swelled demand for air-conditioning, overwhelming dilapidated power grids.

In Baghdad, dozens gathered at the protest hub of Tahrir Square, clashing with police and other security forces.

"Two protesters died this morning. One was shot with a tear gas canister in the head, and another in the neck," a medical source told AFP on Monday.

Their bodies were carried through Tahrir by fellow activists, before being driven to the Shiite holy city of Najaf south of the capital for burial.

"We had no guns, no knives, just our chants," said Ahmad Jabbar, a male protester in the square.

"We (clashed) with them for six hours. They wouldn't even let the ambulances come get the wounded," he told AFP.

- 'No difference' -

More rallies have been called for Monday night, with activists demanding the release of fellow protesters arrested the previous evening.

"If our guys aren't freed, we're going to ramp up our efforts. We're staying in our tents, and we're not afraid," said protester Maytham al-Darraji.

The two deaths were the first since Prime Minister Mustafa al-Kadhemi, who had promised a dialogue with protesters, took office in May.

His office acknowledged "unfortunate events" in protest squares, but insisted security forces had been instructed not to use violence unless absolutely necessary.

It said the government would carry out an investigation and hold those responsible to account.

But protesters were already comparing Kadhemi to his predecessor Adel Abdel Mahdi, who stepped down last year after he was blamed for a violent response to rallies.

"As it turns out, there's no difference between the governments of Kadhemi and Abdel Mahdi," said Darraji.

Around 550 people were killed in the previous wave of rallies and another 30,000 wounded, many of them by military-grade tear gas canisters that can pierce the skull if fired directly rather than lobbed in an arc to disperse crowds.

There was virtually no accountability for those deaths under Abdel Mahdi; Kadhemi had pledged to publish a list of all the victims, carry out investigations and listen to protesters' demands.

Online, Iraqis shared a doctored image of Abdel Mahdi's handover to Kadhemi that depicted the outgoing premier pushing a collection of tear gas canisters and a rifle toward his successor.

The United Nations said it "deplored" the violence.

"Iraqis are in a difficult place facing many challenges. Their right to peaceful protest must be protected unconditionally," the UN's office in Iraq (UNAMI) said.

- 'We want AC!' -

Hundreds of people also staged rallies in the southern cities of Kut and Hillah on Sunday.

In the southern flashpoint city of Nasiriyah, they briefly cut roads and chanted: "We can't stand it, we want AC!"

Kadhemi chaired an emergency meeting alongside Iraq's electricity minister on Monday and said he would "spare no effort" to improve the power situation for citizens.

Protests over power cuts, poor water access and other failing public services are normal across Iraq, whose infrastructure has been battered by decades of war and lack of investment.

Last year, rallies focused on infrastructure morphed into a broader movement slamming the entire ruling class as corrupt, unqualified and beholden to neighbouring Iran.

When Kadhemi came to power, observers saw him as a rare figure of compromise.

But he has struggled to keep Iraq afloat as the coronavirus pandemic has spread, state revenues have been slashed by a collapse in oil prices and security has worsened.

Rockets continue to hit sites across Iraq where foreign troops and diplomats are based, and a German woman was briefly abducted last week.