"People of Izeh, like some other cities, held a protest against economic problems and unfortunately it led to the killing of two people and injuries to some others," Hedayatollah Khademi, the local MP, told the ILNA news agency, adding it was unclear who had fired the shots.

The state broadcaster said two others died in the small western town of Dorud when they were hit by a fire engine stolen by protesters, and ILNA reported a school for clergy and government buildings were torched in the northwestern town of Takestan.

Rouhani finally broke his silence on Sunday night about the protests that mark the biggest test for the regime since mass demonstrations in 2009.

"The people are absolutely free in expressing their criticisms and even protests," Rouhani said in a message on the state broadcaster.

"But criticism is different to violence and destroying public property."

Police used tear gas and water cannon to disperse a small protest in Tehran's Enghelab Square on Sunday evening, according to unverified social media videos.

There were also reports of protests in the cities of Kermanshah, Khorramabad, Shahinshahr and Toyserkan in the west, and Zanjan in north Iran.

The protests began as demonstrations against economic conditions in second city Mashhad on Thursday but quickly turned against the Islamic regime as a whole, with thousands marching in towns across Iran to chants of "Death to the dictator".

Rouhani sought a conciliatory tone, saying that government bodies "should provide space for legal criticism and protest" and calling for greater transparency and a more balanced media.

- 'Not good!' -

US President Donald Trump, a fierce critic of Tehran, said the "big protests" showed people "were getting wise as to how their money and wealth is being stolen and squandered on terrorism".

He accused Iran of "numerous violations of human rights", adding it "has now closed down the internet so that peaceful demonstrators cannot communicate. Not good!"

After initial silence, state media began showing some footage of the demonstrations on Sunday, focusing on young men attacking banks and vehicles, an attack on a town hall in Tehran, and images of a man burning the Iranian flag.

"Those who damage public property, disrupt order and break the law must be responsible for their behaviour and pay the price," Interior Minister Abdolrahman Rahmani Fazli said on Sunday.

Rouhani came to power in 2013 promising to mend the economy and ease social tensions, but high living costs and a 12 percent unemployment rate have left many feeling that progress is too slow.

"Rouhani has run an austerity budget since 2013 with the idea that it's a tough but necessary pill to swallow to manage inflation and currency problems and try to improve Iran's attractiveness for investment," said Esfandyar Batmanghelidj, founder of the Europe-Iran Forum.

"But choosing years of austerity immediately after a very tough period of sanctions is bound to test people's patience," he told AFP.

Since the ruthless repression of the 2009 protests against a disputed presidential election that gave hardliner Mahmoud Ahmadinejad a second term, many middle-class Iranians have abandoned hope of securing change from the streets.

But low-level strikes and demonstrations have continued, with groups such as bus drivers, teachers and factory workers regularly protesting against unpaid wages and poor conditions.