Detectives are in the process of interviewing persons of interest.
Law enforcement sources told KSAT's sister station in Houston, KPRC, they had a person in custody Saturday, almost a week after Barnes was shot to death while riding in a vehicle with her sisters and mother.
The victim's mother and three sisters were also in the auto during the shooting but were not hurt.
Congresswoman Sheila Jackson Lee called for a nationwide manhunt for the suspect, who has been described as a bearded white man, possibly in his forties.
Hundreds of people gathered at a rally Saturday afternoon for the girl near where the shooting happened.
Washington thanked both the celebrities who have raised awareness about the case and those who rallied today, saying that not only were they inspiring to her but also to her children, who are grappling with the death of their sibling.
In an emotional speech, Jazmine's aunt, Sharonica Watt, asked her niece's killer why he took the little girl's life. But investigators have not determined a motive and their focus remains on catching the killer, he said.
'We're asking you to turn yourself in right now, in the name of Jesus.
Jazmine's father, Christopher Cevilla, said his daughter was a loving and caring girl who was 'very smart in school'.
"Investigators do not believe Jazmine's family was the intended target of the shooting", the statement said.
The district said additional counselors will be at the school when students return on Tuesday.
Jazmine's funeral is being paid for by retired National Basketball Association superstar Shaquille O'Neal and veteran Houston Police officer Kenneth Miles.
Washington said she can't remember how many shots were fired but "he emptied out his gun".