The 33-year-old attacker was shot dead at the scene by a police officer who was responding to an unrelated call.
Federal agents are now reviewing social media to look into his beliefs, reports CBS News, the BBC’s US partner.
During the attack, the suspect wore an insignia which has been associated with hate groups.
Six people, including children, were pronounced dead at the scene in the north Dallas suburbs, while two died later in hospital. Three of the injured – ranging in age from 5 to 61 – are still in hospital.
Three members of one family, a young security guard and an engineer from India were among those killed.
The gunman, named by police as Mauricio Garcia, used an AR-15 style rifle and wore combat tactical gear during the shooting. He carried multiple rounds of ammunition.
Witnesses described scenes of panic and horror when the gunman got out of his car near the Allen Premium Outlets mall and began firing on shoppers.
During the attack the killer wore a clothing patch with the letters RWDS, which stands for “Right Wing Death Squad”. This is a phrase popular among right-wing extremists and white supremacy groups.
One line of enquiry is whether he was motivated by these ideals and whether he had links to like-minded people, a law enforcement source told CBS.
A social media page appearing to belong to the gunman also shared extremist views. The profile on a Russian platform reportedly includes posts about mass shootings and white supremacy.
Photos he apparently posted showed Nazi tattoos on his arm and torso. He also shared images last month of a shop near to where the attack took place.
According to the US defence department, the suspect entered the US Army in June 2008 and was “terminated three months later without completing initial entry training” due to “physical or mental conditions”.
The attacker was reportedly working as a security guard at the time of the shooting and did not have a serious criminal record. Officials have searched his parents’ home and a nearby extended-stay motel where he had been recently living.
Speaking to CBS, Steven Spainhouer described how he rushed to the scene after getting a call from his son who reported shooting. He spoke of “unfathomable carnage”.
He said at least three victims could not be saved even after he applied CPR. “The first girl I walked up to was crouched down covering her head in the bushes,” he recalled. “So I felt for a pulse, pulled her head to the side, and she had no face.”
Another witness, Elaine Penicaro, said she was finishing her shopping trip when she heard “all this popping”.
Allen is a racially diverse suburb north of Dallas and has an infamous connection with another recent mass shooting.
A man who lived there in 2019 went on a gun rampage at a Walmart in El Paso, killing 23 people, after posting a racist manifesto online. In February he pleaded guilty to hate crime charges.
The shooting comes days after five people were killed elsewhere in Texas following a dispute with a neighbour. And on Sunday one person was killed and two injured in a shooting on a train in Dallas.
There have been 201 mass shootings this year according to the Gun Violence Archive which defines such incidents as four people injured or killed.