Iranian state TV says the attack is a retaliation after the country’s top commander Qasem Soleimani was killed in a drone strike in Baghdad, on the orders of US President Donald Trump.
The Pentagon says two sites were attacked, in Irbil and Al Asad.
It is unclear if there have been any casualties.
“The president has been briefed and is monitoring the situation closely and consulting with his national security team,” White House spokeswoman Stephanie Grisham said in a statement.
Iran’s Revolutionary Guard said the attack was in retaliation for the death of Soleimani on Friday.
“We are warning all American allies, who gave their bases to its terrorist army, that any territory that is the starting point of aggressive acts against Iran will be targeted,” it said via a statement carried by Iran’s state-run IRNA news agency.
Iranian Foreign Minister Javad Zarif later issued a statement on Twitter, claiming the attack was self-defence and denied seeking to escalate the situation into war.
President Trump tweeted shortly afterwards, insisting “all is well”, while adding that they had not yet assessed possible casualties.