The shooting happened on Saturday at Cielo Vista Mall in El Paso, just a few miles from the US-Mexico border, the BBC reported.
At least 22 people have arrived at hospitals for treatment for their injuries.
One person is in custody, and police did not believe there were multiple gunmen, a spokesperson said, but was unable to provide casualty figures.
"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.
First reports of a shooting emerged at around 11 a.m.
There were reports of gunfire in several locations - in both the Cielo Vista mall and the nearby Walmart, police said - but reports of further gunshots at another nearby mall, Bassett, were not true.
"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.
Police said they no longer believed there was any immediate threat to the public, but were continuing searches. A family reunification area had been set up nearby for those searching for loved ones.
( With inputs from IANS )