"The first thing we have to do is get to the bottom of it and find out to the best of our ability who in fact is responsible, and I think the President has been very clear that we will take appropriate action at a place and time of our choosing," Blinken told the PBS NewsHour later on Wednesday.

The President was briefed Wednesday morning on the attack and had been monitoring details overnight, White House press secretary Jen Psaki said. "We are still assessing the impact of this latest rocket attack, including determining precise attribution," Psaki said Wednesday at a briefing.

Kirby said Iraqi security forces are on scene and investigating.

"We cannot attribute responsibility at this time, and we do not have a complete picture of the extent of the damage. We stand by as needed to assist our Iraqi partners as they investigate," Kirby said.

Kirby said there were "no current reports" of injuries to US service members and said all are accounted for. A damage assessment is underway.

A February 15 rocket attack on coalition forces near the Erbil International Airport in Iraqi Kurdistan killed a civilian contractor and injured nine others, including four American contractors and one member of the US military.

Days later, at least four rockets struck Balad Air Base north of Baghdad. Then on February 22, two rockets landed in Bahgdad's international zone, where many foreign embassies are located. There were no reports of injuries or damage.

One of the groups hit by those strikes, the Iran-backed militia Kata'ib Hezbollah, said in a statement afterward that it had the right to respond to the "barbaric aggression" that killed one of its fighters.

'Vital interests'

US officials had said they hoped the Syria strikes would end attacks on coalition forces. Wednesday's attack could indicate a failure to achieve that goal and that escalation may be ahead.

"One of the things we were certainly hoping to achieve as a result of that strike was to deter future attacks by militia groups on our people, our facilities and our Iraqi partners, and we certainly hope that it has that effect," Kirby told reporters the day before Al-Asad was targeted.

Psaki said Wednesday that the US would reserve the right to respond. "What we won't do is make a hasty or ill-informed decision that further escalates the decision or plays into the hands of our adversaries," she said. "If we assess further response is warranted, we will take action, again, in a manner and time of our choosing. And we reserve that option."

Two US military officials indicated to CNN that as the Biden administration reviews further response options to Wednesday's attack, targets in Syria could be considered because striking in Iraq puts pressure on that government in its own dealings with Iran.

The prospect of military escalation could cause friction for Biden at home, where Democratic lawmakers were critical of the Syria strike.

Sen. Tim Kaine of Virginia said after the strike that offensive military action without congressional approval was "not constitutional absent extraordinary circumstances" and called for Congress to be fully briefed expeditiously.

Rep. Ro Khanna, a California Democrat on the House Foreign Affairs Committee, echoed that call, saying, "We need to extricate from the Middle East, not escalate."

In a speech at the State Department outlining the Biden administration's foreign policy priorities, Blinken said that Biden would put diplomacy before military action, but that "we will never hesitate to use force when American lives and vital interests are at stake."

"That's why President Biden authorized an airstrike last week against Iranian-backed militia groups, targeting US and coalition forces in Iraq," Blinken said. "But in that case, and in future cases when we must take military action we will do so only when the objectives and mission are clear and achievable, consistent with our values and laws and with the informed consent of the American people. And we'll do it together with diplomacy."

'Barbaric aggression'

The rocket launcher used in Wednesday's attack was found in the al-Bayader agricultural area near the town of al-Baghdadi, about 180 kilometers northwest of Baghdad, military sources told CNN. Sabereen news, a pro-Shia militia group, published images on its Telegram page claiming to show the launcher that attacked the base. CNN cannot independently verify the images.

The rockets fired at the Al-Asad base were 122 millimeter, with a longer range and greater firepower than most attacks the US attributes to Iranian-backed militias, according to a US military official directly familiar with initial assessments of the attack. Of the key groups in the area with Iranian support, this type of rocket is most often used by Kata'ib Hezbollah, the official said.

"There is a significant amount of evidence at the scene that has already been evaluated," the official said. The official noted, however, that "more than 75%" of recent attacks have used smaller-scale so-called Katyusha rockets, which are widely available in the area.

Multiple officials said the US views the use of the larger, more powerful 122-millimeter rockets as a "message" from the Iranian-backed groups that they are not intimidated by the recent US airstrike in Syria, the official said.

The US strikes on the Syrian site killed "up to a handful" of militants, a US official told CNN at the time.

Kata'ib Hezbollah released a brief statement the next day, saying the US strikes had killed a fighter "stationed on the Iraqi-Syrian border to protect the land of Iraq and its people from ISIS criminal gangs."

The group called on the Iraqi parliament to expel US forces in the country and prosecute "traitors" who conspire with them, and pledged to respond to the US strikes, saying that despite hopes, the Biden administration was proving to be no different than the Trump administration.

"This barbaric aggression ... indicates without any doubt that the aggressive American policies towards our peoples do not change by changing their administration as the deceived hope, and that its ugly face cannot be hidden by the change of masks," the statement read.

"We affirm the legitimate right of our Iraqi people to respond to this cowardly criminal operation, in revenge for our martyrs," the statement said.

Wednesday's rocket attack comes just two days ahead of a scheduled trip to Iraq by Pope Francis, the first time a Pope has visited the country. The Pope will be staying at the Vatican Embassy throughout his trip, the Vatican said on Tuesday.

This story has been updated with additional developments.