Investigators are regarding 24-year-old Muhammed Youssef Abdulazeez as a "homegrown violent extremist".
However officials said that it was too early to say whether he had been influenced by Islamic militants.
Chattanooga police officers shot and killed Abdulazeez during the attack.
Abdulazeez first shot at a recruiting office from a car and then travelled about seven miles and opened fire on a joint Marine-Navy facility.
After crashing through the gates of the Marine-Navy facility, Abdulazeez fatally wounded one service member inside the building, and the four other Marines were shot and killed outside the building.
All service members appear to have been killed by bullets from the same gun, officials said.
Abdulazeez's motive for the shooting is unclear. Before the attack, Abdulazeez sent a text message linking to a Islamic verse about "declaring war". He also spent several months in Jordan before his death.
Family members have said Abdulazeez suffered from depression and had abused drugs and alcohol.
The FBI's Ed Reinhold said on Wednesday two weapons found at the scene belonged to service members.
It is up to the US military whether those troops were permitted to have those weapons on site.
Mr Reinhold said the whole attack lasted no more than five minutes. The FBI is going to investigate "every possibility and every relative" of Abdulazeez, he said.
Some Marines protected their colleagues during the shootings, officials said.
Twenty Marines and two Navy corpsman were inspecting equipment at the facility at the time of the attack.
Those troops had just returned from a training exercise in California.
"While I cannot share specific details about what happened that morning, our Marines reacted the way you would expect- rapidly going from room to room. They got their fellow Marines to safety," said Maj Gen Paul Brier of the 4th Marine Division.
Troops "doubtlessly saved lives", Mr Reinhold said.
The New York Times reported that some of the Marines who were shot to death "diverted" Abdulazeez away from a group of potential victims.
An official told the newspaper that the attacks "could have been a lot worse- a horrible, horrible massacre".
Authorities have determined that a gunman who killed five US service members when he attacked two military facilities in Chattanooga, Tennessee, acted alone.