Three days after a Saudi air force lieutenant shot dead three people at a naval base in Pensacola, Florida, the FBI said it was working “with the presumption that this was an act of terrorism”.

The gunman who struck in a classroom at Naval Air Station Pensacola on Friday morning was killed by sheriff’s deputies, two of whom were among eight people wounded. All were expected to survive.

No arrests had been made, the FBI agent in charge said at a press conference on Sunday, adding that all international students at the base have been accounted for.

“We are, as we do in most active-shooter investigations, working with the presumption that this was an act of terrorism,” said Rachel Rojas.

“There are a number of Saudi students who are close to the shooter and continue to cooperate in this investigation. The Saudi government has pledged to fully cooperate with our investigation.”

Saudi state media has reported that the Crown Prince Mohammed bin Salman called President Trump expressing his condolences for the families affected by the shooting.

The crown prince also assured Trump that Saudi authorities would offer their absolute cooperation with the US and provide all information that would help the investigations.

The defense secretary, Mark Esper, said he had asked the Pentagon to review screening procedures for soldiers from other countries sent to the US for training. Key allies of Donald Trump called for Saudi Arabia to be suspended from such programs.

The FBI identified the gunman as Mohammed Alshamrani, 21.

A US official told the AP investigators believe Alshamrani visited New York City, including Rockefeller Center, days before the shooting. The FBI was attempting to question anyone who may have interacted with him, the official said.

Authorities also believe the gunman made social media posts about US support for Israel and alleging Americans were anti-Muslim, the official said. Investigators were continuing to examine his online profiles and social media postings.

On Saturday, a US official told the Associated Press the Saudi gunman hosted a dinner party this week where he and three others watched videos of mass shootings. The official said one of the three students who attended the party then recorded video outside the classroom building while the shooting took place, while two Saudi students watched from a car.

Rojas said the FBI was “looking very hard at uncovering his motive and I would ask for patience so we can get this right”. She added that 80 FBI special agents, 100 support staff and scores of other investigators from the navy and multiple federal agencies were working the case.

There’s something really bad here. We need to slow this program down and reevaluate Lindsey Graham

Rojas said the pistol used – a Glock 9mm handgun that can be paired with a magazine holding 33 rounds – was legally purchased somewhere in Florida. According to US regulations, it is legal for a foreigner on a non-immigrant visa to buy a gun if certain conditions are met, including if they simply have a hunting license.

Esper appeared on Fox News Sunday. He said he had asked top defense officials to “make sure we’re taking all necessary precautions” to make sure military installations are secure.

The shooting was the second at a US naval base in a week. A sailor whose submarine was docked at Pearl Harbor, Hawaii, opened fire on three civilian employees on Wednesday, killing two before killing himself.

Esper also said he had asked the Pentagon to “review what our screening procedures are” before soldiers from other countries come to the US for training programs he said were “very important to our national security”.

More than 850 Saudis are in the US for training activities, among more than 5,000 foreign students from 153 countries.

“This has been done for many decades,” Trump told reporters on Saturday. “I guess we’re going to have to look into the whole procedure. We’ll start that immediately.”

Two leading Trump allies in Congress said Saudi Arabia should be suspended from US training programs.

“We need to suspend the Saudi program until we find out what happened here,” the South Carolina senator Lindsey Graham told Fox News’ Sunday Morning Futures with Maria Bartiromo.

Graham said he liked the idea of training foreign pilots “but there’s something really bad here. We need to slow this program down and re-evaluate.”

Matt Gaetz, a Florida congressman whose district includes the base in Pensacola, told ABC’s This Week the shooting “has to inform our ongoing relationship with Saudi Arabia”.

Saying he and Trump had been communicating regularly, he added: “We should not be taking new incoming Saudi students until we’re absolutely confident in our vetting process.”

The navy identified the men who died and hailed them as heroes for trying to stop the gunman and flagging down first responders after being shot.

Capt Tim Kinsella, commanding officer of Naval Air Station Pensacola, said in a statement: “When confronted, they didn’t run from danger; they ran towards it and saved lives.”

One of the victims was a 23-year-old graduate of the US Naval Academy who alerted first responders to where the gunman was after he had been shot several times.

“Joshua Kaleb Watson saved countless lives today with his own,” his older brother, Adam Watson, wrote on Facebook. “He died a hero and we are beyond proud.”

Airman Mohammed Sameh Haitham, 19, of St Petersburg, Florida, joined the navy after graduating high school last year, according to the Tampa Bay Times.

Haitham’s mother, Evelyn Brady, herself a navy veteran, said the commander of her son’s school called her and told her Haitham had tried to stop the gunman. The former track and field star had been assigned to flight crew training and was looking forward to graduating from the program later this month, she said.

“He said he was going to get his flight jacket for Christmas,” she said. “Now that’s not going to happen.”

The third victim was airman apprentice Cameron Scott Walters, 21, of Richmond Hill, Georgia.