A reformist website said a nephew of Iranian opposition leader Mir Hossein Mousavi was killed in clashes between protesters and security forces in Tehran on Sunday.
Another opposition website said Sunday evening at least four protesters were killed in the northwestern city of Tabriz on Sunday during clashes between opposition supporters and security forces. "During clashes between security forces and protesters ... at least four protesters were killed in Tabriz and many others wounded," said the Jaras website, which earlier reported four protesters were killed in Tehran.
The White House condemned what it called the "unjust suppression" of civilians by the Iranian government and said the United States was on the side of protesters.
"We strongly condemn the violent and unjust suppression of civilians in Iran seeking to exercise their universal rights," White House National Security Council spokesman Mike Hammer said in a statement.
"Hope and history are on the side of those who peacefully seek their universal rights, and so is the United States," Hammer said.
"Governing through fear and violence is never just, and as President (Barack) Obama said in Oslo -- it is telling when governments fear the aspirations of their own people more than the power of any other nation," Hammer said in a reference to Obama's speech this month accepting the Nobel Peace Prize.
The Parlemannews website said Ali Mousavi, 20, was killed in clashes on Sunday and his body had been taken to a hospital.
Jaras said unrest also spread to other parts of Iran, including the holy city of Qom, in reports that could not be independently verified.
The events underlined escalating tension in the Islamic Republic six months after a disputed presidential poll plunged the oil producer into turmoil and exposed widening splits within the clerical and political establishment.
Jaras said police shot dead three protesters in central Tehran. It later said a fourth demonstrator was also killed in clashes in the capital, without giving details.
"Three people were killed and two others were wounded when police opened fire at protesters," the website said.
Any such violent incidents could provoke further opposition protests.
"We will kill those who killed our brothers," Jaras quoted demonstrators as chanting.
These were the first reported killings in street protests since widespread unrest and violence in the immediate aftermath of the June poll in which the opposition says more than 70 people died.
The authorities have estimated the post-vote death toll at about half that number, including pro-government militiamen.
Tehran police chief Azizollah Rajabzadeh, speaking about Sunday's protests, said: "So far there have been no reports of killings and no one has been killed up to now," according to the ISNA news agency. He said some arrests had been made.
Tens of thousands of demonstrators had packed the streets of Tehran and clashes also erupted in the cities of Shiraz, Isfahan, Najafabad, Mashhad and Babol, Jaras said.
It said 20 people were detained in Qom and Mashhad and that protests would continue in Tehran on Sunday evening. Shots were heard in northern Tehran after nightfall.
English-language state television reported sporadic clashes in Tehran and said a bank and bus stop were set ablaze, showing pictures of protesters and fires with thick smoke. It said police had fired into the air to disperse demonstrators.
The official IRNA news agency said two women and a child were hurt when rioters threw stones at people marking Ashura. It is one of the main Shiite holy days when the faithful commemorate the slaying of the Prophet Mohammad's grandson Hussein in Kerbala in present-day Iraq in 680 AD.
The semi-official Fars News Agency said supporters of opposition leader Mousavi "followed the call of the foreign media" and took to the streets - a reference to the government position that the unrest is being stoked by foreign enemies of the Islamic Republic.
It said the group of "deceived hooligans" damaged public and private property and "disrespected" the holy Shiite day of Ashura, without elaborating.
Foreign media have been banned from reporting directly from opposition demonstrations since the June election.
Despite scores of arrests and security crackdowns, opposition protests have flared repeatedly since the June poll, which the opposition says was rigged to secure hardline President Mahmoud Ahmadinejad's re-election.
Reformist websites said there had also been clashes in Tehran on Saturday, with baton-wielding riot police firing tear gas and warning shots to disperse Mousavi supporters.
The authorities had warned the opposition against using the two-day Shiite Muslim Tasoua and Ashura festival on December 26-27 to revive protests against the clerical establishment.
"The Iranian nation has shown tolerance so far but they should know that the ... system's patience has a limit," Mojtaba Zolnour, a representative of Supreme Leader Ayatollah Ali Khamenei in the Revolutionary Guards, said, Fars reported.
This year's Ashura on Sunday coincided with the traditional seventh day of mourning for leading dissident cleric Grand Ayatollah Hossein Ali Montazeri, who died a week ago at the age of 87 in the holy Shiite city of Qom.
A spiritual patron of the movement of opposition leader Mousavi, he was a fierce critic of the hardline clerical establishment.
The unrest that erupted after the June vote is the biggest in the Islamic state's 30-year history. Authorities deny opposition charges that voting was rigged.
The turmoil has complicated the international dispute over Iran's nuclear program, which the West believes may have military ends, not just civilian purposes. World powers have set an end-of-year deadline for Iran to agree a UN-drafted deal to ship most of its low-enriched uranium abroad in exchange for fuel for a Tehran research reactor.
It has also set back tentative US moves towards a rapprochement with Iran initiated by US President Barack Obama when he took office in January.