One of Iran's most senior officials appears to have posted a direct message to US President Donald Trump after an attack on US bases in Iraq today.

Iran's top nuclear negotiator, Saeed Jalili, tweeted a lone image of the Iranian flag shortly after a series of surface-to-surface missiles struck at least two military bases where US and coalition troops are based.

It might not seem like much, but the image — retweeted more than 4900 times already — mirrors a tweet by the US President following the assassination last week of Iranian military commander Qassem Soleimani.

Trump posted a picture of the US flag to his Twitter feed on January 3.

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Iranian officials said the attacks, confirmed by US officials, marked the beginning of a promised retaliation against the US for the killing of top Iranian military commander Qassem Soleimani.

"The fierce revenge by the Revolutionary Guards has begun," Iran's Islamic Revolutionary Guard Corps said in a statement on a Telegram channel.

The Revolutionary Guard later warned that if the US responds to its attack in Iraq, Iran "will respond to you in America".

US Defence Department spokesman Jonathan Hoffman said the rockets were clearly from Iran and the attacks were "targeted" at the bases in Ain Assad and Irbil.

Hoffman said the US was "working on initial battle damage assessments". There was no immediate word on injuries.

Iran's Revolutionary Guard said the Ain Assad air base was "completely destroyed".

The attack was "a total success by all accounts," the Revolutionary Guard said.

Iranian state TV said the attack was a revenge attack for the killing of Soleimani, whose funeral on Tuesday in his hometown of Kerman prompted angry calls to avenge his death, which drastically raised tensions in the Middle East.

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State TV said the operation's name was "Martyr Soleimani."

It said the Guard's aerospace division that controls Iran's missile program launched the attack. Iran said it would release more information later.