Seyyed Ali Mousavi, Mousavi's 35-year-old nephew, was among the eight people killed, with the Parlemannews website saying he was shot during clashes at Tehran's Enghelab square "and was martyred after he was taken to Ebnesina hospital".
State television attributed his death to "unknown assailants".
Norooz, an opposition website, said police had fired teargas to disperse supporters of Mousavi outside the hospital.
Opposition reports that police fired on protesters during the demonstrations, could not be independently verified as foreign media face severe restrictions in Iran.
Ahmad Reza Radan, Iran's deputy police chief, acknowledged that "several people" had been killed, but denied that police had used guns to contain the protests.
Referring to four of the deaths, Radan said: "One fell off a bridge, two died in car accidents and one was killed by a bullet."
"As the police was not using firearms this [death] is suspicious and it is being investigated," he said.
Opposition leaders criticised the government for the killings in what are some of the bloodiest confrontations since the demonstrations that followed the disputed June 12 presidential poll won by Mahmoud Ahmadinejad.
"What has happened to this religious system that it orders the killing of innocent people during the holy day of Ashoura?" Mehdi Karroubi, one of the defeated reformist candidates in the election, said in a statement reported on the Jaras website.
The website also said that unrest had spread to other parts of Iran, including the holy city of Qom, Shiraz, Isfahan, Najafabad, Mashhad and Babol.
Security forces attacked
But others played down the protests. Kian Mokhtari, a columnist and political commentator in Tehran, told Al Jazeera that reports of the protests had been exaggerated.
"There are 10 million Basji [religious security forces] in Iran and if the Supreme Leader intended to crack down - you can't call this a crackdown," he said.
"You have people coming out on to the streets and damaging public property and they get stopped by police."
He said: "Peaceful demonstrations were planned for today [Monday], so I have no idea why these several thousand-at-most people went out and decided to pick a fight.
"But they went out and their first act was to set fire to a bank and then they attacked the Ashoura mourners.
" Then a fight ensued between the two element and the riot police came in. The riot police were badly beaten and bruised."
Payam Akhavan, who co-founded the Iran Human Rights Documentation Centre, characterised the protests as a result of Iran's disputed June election, which saw Mousavi and others beaten to the presidency by Ahmadinejad, the incumbent.
"The regime has tried to portray the Green [opposition] Movement, the protests, as a very limited segment of Iranian society - middle class, secular students in Tehran," he told Al Jazeera.
"But now it's clear that the protests against the regime encompass a very broad cross-section of Iranian society, including devout Muslims."
Mohammad Marandi, a professor at Tehran University, said that the protests were being organised from abroad, a sentiment frequently stressed by the Iranian government.
"They are largely being orchestrated from abroad, they are orchestrated from television networks that are being beamed in from the United States and Europe," Marandi told Al Jazeera.
|The protests are some of the worst since those following the disputed presidential poll [EPA]
"The important thing is that initially the protests against Mr Ahamdinejad were more homegrown ... but gradually the pro-American opposition, the opposition supported from outside the country, began to have a greater role."
But Trita Parsi, the president of the National Iranian American Council, an advocacy group, rejected that assessment.
"This is a movement that is homegrown, that is finding its inspiration from what is taking place inside the country," he told Al Jazeera.
"There is no evidence or any clear indication that they will be taking any particular orders or money from outside."
Iran's government has been criticised internationally for its clampdown on the protests.
In Washington, the White House strongly condemned the "violent and unjust suppression" of civilians.
Angela Merkel, the German chancellor, said the actions of the security forces were "unacceptable".