The nephew of Iran's opposition leader Mir Hossein Mousavi was threatened days before he was killed in Sunday's protests, a prominent supporter says, BBC reported.
Mohsen Makhmalbaf, a film-maker based in Paris, told the BBC that secret police called Seyed Ali Mousavi several times, saying: "We will kill you."
But a police statement reportedly said Mr Mousavi was killed by "terrorists".
Meanwhile a regional representative of Iran's supreme leader said opposition leaders should be executed.
"Those who are behind the current sedition in the country ... are mohareb [enemies of God] and the law is very clear about punishment of a mohareb," state media quoted Ayatollah Abbas Vaez-Tabasi, an influential figure in Khorasan province, as saying.
The sentence for mohareb under Iran's Sharia law is death.
Earlier Iranian President Mahmoud Ahmadinejad accused Western countries of organising the protests.
Tehran has rejected international calls for it to halt the crackdown. State media say at least eight people died in the protests.
The Irna news agency quoted Mr Ahmadinejad as describing the opposition rallies as a "nauseating masquerade".
"Iranians have seen lots of these games," he said.
"Americans and Zionists are the sole audience of a play they have commissioned and sold out."
'Isolated and monitored'Mr Makhmalbaf said the alleged death threats against Seyed Ali Mousavi were part of the government's continuing attempts to weaken the opposition.
"During last week of his life, several times, the Iranian secret police call him and inform him that 'we will kill you' and they did," he told the BBC World Service's Newshour programme, before describing his death.
"From one car, five persons from secret police came in front of him and shot at his chest and before he arrived at hospital he was dead."
A police statement, quoted by AFP news agency, said Mr Mousavi had been shot by "people aboard a vehicle".
But it added that "an extensive investigation is ongoing to identify the terrorists behind the incident".
Mr Makhmalbaf said Mr Mousavi's uncle was being isolated and closely monitored by the authorities.
"He's in a very bad situation," he said.
"The police have arrested all of his assistants and he's under control in everything. He can't speak by phone with his friends. He's using pen and paper to [communicate], even in his house.
"He is in his house. He can come out, but not freely because the bodyguards for him are from the Revolutionary Guard."
Meanwhile the US-based son of another opposition leader, Ebrahim Yazdi, said his father's arrest would not hinder attempts to bring democracy to Iran.
"The government is using the tools of fear and intimidation, but the key is not to allow that to slow anybody down," Youseph Yazdi said in an interview for US National Public Radio.
On Monday, state-owned English-language Press TV said eight people had died. State TV had earlier reported that at least 15 people were killed.
The official death toll is the highest since June, when mass protests were sparked by President Ahmadinejad's disputed re-election.
Iranian security forces have been on alert since the influential dissident cleric, Grand Ayatollah Hoseyn Ali Montazeri, died earlier this month.
His funeral attracted tens of thousands of opposition supporters, many of whom shouted anti-government slogans.