The man who shot and killed five US troops in Tennessee suffered from depression, his family says.

Four Marines and a sailor were killed in the attack – which authorities are treating as an “act of terrorism” – before the gunman, Mohammad Youssuf Abdulazeez, died in a shoot-out with police.

“There are no words to describe our shock, horror and grief,” Abdulazeez’s family said in a statement cited by an NPR public radio journalist.

• US soldiers killed by gunman

“The person who committed this horrible crime was not the son we knew and loved,” they added.

“For many years, our son suffered from depression. It grieves us beyond belief to know that his pain found its expression in this heinous act of violence.”

Expressing its condolences to the families of the victims, the Abdulazeez family also vowed to continue to co-operate with law enforcement.

Investigators are seeking to determine a motive behind Thursday’s rampage on two military centres in Chattanooga.

The Federal Bureau of Investigation has asked foreign intelligence agencies to help trace Abdulazeez’s movements and activities abroad, and analysts are monitoring his activity on social media.

A law enforcement official told the New York Times that investigators were looking into a text message Abdulazeez allegedly sent to a friend hours before the shooting.

The text reportedly included an “Whosoever shows enmity to a friend of Mine, then I have declared war against him,” according to the newspaper.

Abdulazeez, 24, was a naturalised US citizen born in Kuwait.

“Every one of our resources are being devoted to this investigation,” Chattanooga Mayor Andy Berke told CNN.

But the FBI warned against jumping to conclusions, after Michael McCaul, chairman of the House of Representatives Homeland Security Committee, branded the assault “an ISIS-inspired attack,” using an alternative acronym for the Islamic State group.

“At this time, we have no indication that he was inspired by or directed by anyone other than himself,” FBI special agent Ed Reinhold said, referring to Abdulazeez.

The shooting has jarred the city of 168,000, where the Islamic Society of Greater Chattanooga condemned the attack and cancelled its end-of-Ramadan Eid al-Fitr celebrations out of respect for the victims.

The four Marines died on Thursday, while the fifth victim, a sailor, succumbed on Saturday to injuries he sustained in the assault.

The US Navy named the sailor as Logistics Specialist 2nd Class Randall Smith, a father of three daughters who had recently re-enlisted and transferred to Chattanooga.

“It’s hard to understand how somebody can hurt somebody that’s serving for you, for your freedom, for your safety,” his step-grandmother Darlene Proxmire told WANE television in Indiana.