Four people are thought to have died in Tehran and another four in Tibriz, according to several opposition websites.
However this figure has been disputed by police, who said 300 people had been arrested and five people had been killed by unknown assailants.
A police statement said security forces would investigate what were described as 'suspicious deaths.'
It has been confirmed that a nephew of opposition leader Mir Hossein Mousavi was among those killed in Tehran.
Reformist website Parlemannews said 35-year-old Seyed Ali Mousavi died in hospital after being shot in the chest at Enghelab Square.
Foreign media have been banned from reporting directly from opposition demonstrations after Iran's disputed June election.
State-sponsored and opposition websites - as well as sites such as Twitter - have provided an outlet for information, much of it unverified, on the protests.
Police, according to witnesses, had first used batons and tear gas in the crackdown, which followed stern warnings by the authorities that they would crush attempts to use the Ashura processions as a launchpad for protests.
Opposition website Jaras said clashes continued in Tehran overnight and police have been firing tear gas to disperse protestors in some squares.
A witness told Reuters there was a heavy presence of both security forces and opposition backers in central Tehran.
Rahesabz.net reported three unidentified deaths on its website. 'Three of our compatriots were martyred and two were injured in clashes. The reporter who was on the scene said these three were directly shot at by military forces.'
This year's religious holy day of Ashura falls on Sunday, the seventh day of mourning for Montazeri.
In the Shia religion, friends and family traditionally gather to commemorate the deceased a week after their death.
Ashura is one of the main Shia holy days, when the Shia faithful commemorate the slaying of the prophet Mohammad's grandson Hussein in Kerbala in 680 AD.
Authorities had warned the opposition not to use the festival to revive protests.
'The Iranian nation has shown tolerance so far but they should know that the ... system's patience has a limit,' said Mojtaba Zolnour, a representative of Supreme Leader Ayatollah Ali Khamenei, the semi-official Fars News Agency reported.
The statement said Mousavi supporters had 'followed the call of the foreign media'.
