The White House on Wednesday demanded that Iran release demonstrators rounded up in countrywide protests, raising pressure on Tehran as US President Donald Trump weighs the future of a key nuclear deal.

"The Trump Administration is deeply concerned by reports that the Iranian regime has imprisoned thousands of ‎Iranian citizens in the past week for engaging in peaceful protests," the White House said in a statement.

It added that reports of some demonstrators being "tortured or killed... while in detention are even more disturbing," and slammed Tehran for what it called its "true brutal nature."

"We will not remain silent as the Iranian dictatorship represses the basic rights of its citizens and will hold Iran's leaders accountable for any violations," White House press secretary Sarah Sanders said.

"The United States calls for the immediate release of all political prisoners in Iran, including the victims of the most recent crackdown," the statement read.

Violent unrest fuelled by economic grievances erupted in dozens of Iranian cities between December 28 and January 1, leaving at least 21 people dead.

Various figures have come from official sources in Iran about the number of people arrested, with reformist lawmaker Mahmoud Sadeghi putting the total at 3,700.

Officials have said the majority have since been released, with only the main "instigators" facing trial.

An investigation has been opened into the death in custody of a young Iranian in Tehran's infamous Evin prison, the country's judiciary said Tuesday, with Sadeghi linking the death to recent protests.

A reluctant Trump is expected on Friday to waive a series of sanctions against Iran, as part of a deal to curb the Islamic republic's nuclear program.