President Donald Trump says Tehran has no plan to execute protesters, but still hasn't ruled out US military action against the country over its violent crackdown on anti-government demonstrators.

The US and UK are both reducing the number of personnel at the Al-Udeid air base in Qatar, while officials have told CBS, the BBC's US partner, that a partial American withdrawal is a precautionary measure.

Iran's airspace was closed to nearly all flights for five hours overnight, with several airlines announcing that they will reroute flights around Iran.

The Foreign Office has also temporarily closed the British embassy in Tehran, which will now operate remotely, a government spokesperson said.

Speaking from the White House on Wednesday, Trump said his administration had been told on good authority that the killing in Iran is stopping, and there's no plan for executions.

When questioned by a reporter, Trump said that very important sources on the other side had informed him of the developments, adding that he hoped the reports were true.

Initial demonstrations over the collapse of the Iranian currency began in late December, but swiftly expanded into a wider crisis of legitimacy for Iran's clerical leadership.

Trump had earlier threatened very strong action against Iran if the government executed protesters, after reports emerged that a 26-year-old man who had been arrested during the height of the demonstrations had been sentenced to death.

Erfan Soltani was scheduled to be executed on Wednesday, his family had told BBC Persian. They later told the Norway-based Kurdish human rights group Hengaw that his execution had been postponed.

Iranian authorities have reportedly failed to give Soltani's family any more information about his case, citing only that he had been arrested in connection with a protest.

Responding to reports of potential executions, Iran's Foreign Minister Abbas Araghchi said hanging is out of the question and there would be no hanging today or tomorrow.

In addition to the temporary closure of the British embassy in Tehran, the US Mission to Saudi Arabia has advised its personnel and citizens to exercise increased caution and limit non-essential travel to any military installations in the region.

Trump has been reluctant to lend his support to any challengers to Iran's Supreme Leader, Ayatollah Ali Khamenei, saying that we aren't really up to that point yet in an interview with Reuters on Wednesday.

Many Iranian demonstrators called for Pahlavi's return during the protests which began more than three weeks ago.

Tehran has imposed an internet blackout on the country since last Thursday, as authorities stepped up their crackdown on anti-government demonstrations.

The project Human Rights Activists News Agency (HRANA) has reported at least 2,435 confirmed deaths among protesters, including 13 children, with more than 882 further incidents still under investigation.