Three law enforcement officials who spoke to the Associated Press named the gunman as Mauricio Garcia, who was fatally shot by a police officer who happened to be near the suburban Dallas mall.
The officials spoke on condition of anonymity in order to discuss details of an ongoing investigation.
One of the officials said Garcia had been staying at a nearby motel.
The official said investigators have been searching the motel and a home in the Dallas area connected to the suspect.
The official said police also found multiple weapons at the scene after Garcia was fatally shot by a police officer.
The weapons included an AR-15-style rifle and a handgun, according to the official.
The gunman's name emerged as the community of Allen mourned for the dead and awaited word on the seven people who were wounded.
John Mark Caton, senior pastor at Cottonwood Creek Church offered prayers during his weekly service for victims, first responders and the shoppers and employees who "walked out past things they never should have seen".
"Some of our people were there. Some perhaps in this room. Some of our students were working in those stores and will be changed forever by this," Caton said.
Recalling phone conversations with police officers, he said: "There wasn't an officer that I talked to yesterday that at some point in the call didn't cry".
Police did not immediately provide details about the victims at Allen Premium Outlets, a sprawling outdoor shopping centre, but witnesses reported seeing children among them.
Some said they also saw what appeared to be a police officer and a mall security guard unconscious on the ground.
A 16-year-old pretzel stand employee, Maxwell Gum, described a virtual stampede of shoppers.
He and others sheltered in a storage room.
"We started running. Kids were getting trampled," Gum said.
"My co-worker picked up a four-year-old girl and gave her to her parents."
Dashcam video circulating online showed the gunman getting out of a car and shooting at people on the sidewalk.
More than three dozen shots could be heard as the vehicle that was recording the video drove off.
Allen Fire Chief Jonathan Boyd said seven people, including the shooter, died at the scene.
Two other people died at hospitals.
Seven people remained hospitalised on Sunday - three in critical condition and four in fair condition, the Allen Police Department said in a statement.
An Allen police officer was in the area on an unrelated call when he heard shots at 3.36pm, the department wrote on Facebook.
"The officer engaged the suspect and neutralised the threat. He then called for emergency personnel," the post said.
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