The shooting happened at Cielo Vista Mall in El Paso, just a few miles from the US-Mexico border.
At least 22 people arrived at local hospitals for treatment for their injuries, the report say
"We don't want to do anything prematurely until we have accurate facts for you," Sergeant Robert Gomez said, declining to provide details on the injured.
"Initial reports of the weapon was a rifle - I can't confirm that yet," the spokesman said.
He could not confirm any details about the suspect, except that he was male. No officers had fired their weapons in apprehending him, he said.
"This is a tragedy that we never, ever would have thought would happen in El Paso," El Paso Mayor Dee Margo told CNN. "It tears me up."
Texas state Governor Greg Abbott said the attack was "a heinous and senseless act of violence".
"We have deployed troopers, special agents, Texas Rangers, tactical teams, and aircraft to the scene in a support role," he said. "The state of Texas will do everything it can to ensure justice is delivered to the perpetrators".
President Donald Trump called the shooting, “terrible” on Twitter, saying that he offered the government's support to Texas Governor.
“Working with State and Local authorities, and Law Enforcement,” the President wrote. "Spoke to Governor to pledge total support of Federal Government. God be with you all!”
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