But they are investigating it as a potential "lone wolf" terror attack, the official said.
A man wielding a machete stormed the restaurant
Thursday night and attacked several people before police killed him following a car chase, authorities said.
His motive remains unclear, but investigators are trying to determine if the Nazareth Restaurant & Deli was chosen under the mistaken belief the restaurant had a Jewish owner, sources told CNN.
The owner told The Columbus Dispatch newspaper
that the attacker came in half an hour before the incident and asked about him.
The owner is an Israeli Christian-Arab.
"I'm the minor, minor, minor minority of the minorities. So nobody likes me anyways, but thank God we are in America," Nazareth owner Hany Baransi told the paper.
The FBI is assisting in the investigation.
Four people were injured during the attack and taken to Grant Medical Center, Police Chief Kim Jacobs said in a Friday afternoon press briefing.
A 54-year-old man was in critical but stable condition and a man and woman, both 43, were listed in stable condition, she said.
Another 43-year-old man was treated and released from the hospital, she said.
Columbus police identified the slain suspect as Mohamed Barry, 30, of Columbus. He was a Somali native with a criminal history that includes a history of drug use, a law enforcement official told CNN.
Jacobs said she didn't know Barry's motive or whether Columbus police had dealt with him in the past. She confirmed there appear to be no accomplices.
When reporters pressed her on whether this was an act of terror, Jacobs deferred questions to federal authorities.
Nobody inside the restaurant said they knew Barry, according to police.
After the suspect returned to the restaurant, he attacked a man and woman at a booth just inside the door, Columbus police Sgt. Rich Weiner said.
He "immediately began swinging a machete at customers and employees inside," Weiner said. "There was no rhyme or reason as to who he was going after as soon as he walked in, according to some of the witnesses."
Karen Bass was inside the restaurant when it all happened.
"He came through the front door and then started just systematically hitting people," she told CNN affiliate WSYX-TV
. "I thought it was a personal thing, and then he just started down the row hitting everybody."
Two customers ran out and called 911 while another engaged the suspect in a struggle. Some customers threw chairs, then an employee "grabbed something from behind the register" and stopped the attack, according to Weiner.
The man fled, but officers later pulled him over in a vehicle. He eventually emerged from the vehicle with a machete and another knife in his hands.
According to Weiner, the man then "lunged across the hood at the officers."
"Another officer in a cruiser fired a couple shots at him and put him down," Weiner said.
Jacobs said the officer who shot Barry was a 25-year veteran.
The FBI is helping local police with the investigation, which includes trying to determine the attacker's motive, a U.S. law enforcement official said on condition of anonymity.