The attacker, an Iraq war veteran, was known to the FBI after walking into their offices in Anchorage in November last year and claimed he was being forced to fight for ISIS, CBS said.
Authorities have refused to rule out terrorism as the motive for the attack.
The lone shooter, who was wearing a Star Wars T-shirt, is understood to have flown into Fort Lauderdale Airport in Florida on flight Delta 2182 for Alaska with a checked gun, according to reports.
After arriving in baggage reclaim, he pulled his weapon from his bag and moved to the bathroom to load it.
He remerged soon afterwards and started randomly firing shots out of what is believed to be a 9mm handgun at passengers near a baggage carousel, killing five.
The city's mayor said emergency services were called to the scene just before 1pm local time (6pm UK time).
Florida Senator Bill Nelson reportedly said the gunman has been identified as 26-year-old Esteban Santiago, who was carrying military ID.
Fort Lauderdale Airport in Florida
Fri, January 6, 2017Scenes from the Fort Lauderdale Airport in Florida
NBC
He is reportedly from New Jersey and lived in Alaska for some time. It is believed he served in the Army National Guard until six months ago where he was discharged for poor performance.
The suspect’s family have claimed Santiago had been admitted to hospital for psychiatric problems and that his tour of Iraq had changed his mindset.
The first picture of the alleged shooter has emerged, showing a young man holding a baby. CBS New York have reported that the family confirmed it is Santiago pictured.
Broward County commissioner Chip LaMarca wrote on Facebook: "The shooter was a passenger on a Canadian flight with a checked gun.
"He claimed his bag and took the gun from baggage and went into the bathroom to load it. Came out shooting people in baggage claim."
CBS
However an official from the the Canadian embassy said the suspect did not fly from Canada and was not on the Canadian flight.
Air Canada later tweeted: "We confirm that we have no record of such a passenger by that name, or checked guns, on any of our flights to Fort Lauderdale."
The shooter, who said nothing, laid down on the ground after running out of ammunition where he was then arrested by police.
He lay on the ground and surrendered to police.
Officers then asked everyone not to move and the suspect was led away. No more details on the attacker have yet been released by authorities.
Around 90 minutes after the attack, panic broke broke out at another terminal with passengers and police running frantically.
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