The FBI says it is investigating the attack where three people were shot dead at a Florida US navy base on Friday "with the presumption that it was an act of terrorism".
A Saudi Air Force lieutenant killed three people and wounded eight at the base in Pensacola on Friday morning, before being fatally shot by police.
The FBI said on Sunday it was likely the suspect acted alone.
Rachel Rojas, special agent in charge of the FBI’s Jacksonville office, said on Sunday that the shooter used a Glock model 45 9mm handgun that he had bought legally in Florida.
The suspect, identified by the FBI as 2nd Lt Mohammed Alshamrani, 21, bought the gun from a shop this year, a police source told CNN.
Alshamrani obtained a hunting licence, which allows someone on a non-immigrant visa to buy a gun, the source said.
“We currently assess there was one gunman who perpetrated this attack and no arrests have been made in this case,” said Ms Rojas, the lead investigator in the case.
“We are looking very hard at uncovering his motive and I would ask for patience so we can get this right."
Authorities confirmed the killer was a member of the Royal Saudi Air Force who was on the base as part of a US Navy training programme designed to foster links with allies.
Ms Rojas declined to comment on the shooter’s travels while he was based in Pensacola.
The three victims of the attack have been identified as Cameron Scott Walters, 21, Joshua Kaleb Watson, 23, and Mohammed Sameh Haitham, 19.
The Navy on Saturday praised the three sailors for their "exceptional heroism and bravery in the face of evil".
A sheriff’s deputy fatally shot the gunman, authorities said, ending the second deadly attack at a US military base in a week.
Within hours, Saudi Arabia’s King Salman had called US President Donald Trump to extend his condolences and pledge his kingdom’s support in the investigation.
Ms Rojas said there were several Saudi students who were close to the shooter and were co-operating with investigators.
“Their Saudi commanding officer has restricted them to base and the Saudi government has pledged to fully co-operate with our investigation,” she said.
“I thank the kingdom for their pledge of full and complete co-operation.”