The FBI says it's still unclear if the man behind a shooting rampage at two military bases in Tennessee had been radicalised or was just mentally ill.

Four US marines and a sailor were killed when 24-year-old Mohammad Youssuf Abdulazeez, opened fire on two military centres in Chattanooga before being shot dead by police on Thursday.

Authorities have said they are treating the case as an "act of terrorism", confirming that he acted alone.

"At this time we are treating him as a homegrown violent extremist," FBI special agent Ed Reinhold told reporters.

"We believe he acted on his own on that day."

Nearly 1000 FBI agents are pursuing hundreds of leads as they try to understand what motivated Abdulazeez, a naturalised US citizen who was born in Kuwait.

His family has spoken of a history of mental illness and drug abuse and said they sent Abdulazeez to stay with family in Jordan in hopes of getting his life back on track.

Investigators are examining his overseas travel and his online history to see if he came into contact with extremists who may have encouraged him to carry out the attack.

"At this point it's too early in the investigation to determine whether or not he had been radicalised," Reinhold said.

"We are pursuing that as a possibly but it's too early to tell."

Reinhold urged the media to be "responsible when talking about the case" and "not to jump to conclusions about what happened."