Bates’ daughters, Lilianais Griffith, 14, and Luluvioletta Bandera-Magret, 7, were nowhere to be found. The girls lived with their mother and a roommate, Terry Allen Miles.
Miles, 44, is suspected of abducting the girls, Banks said.
"At this point, we believe that the two daughters are in the presence of Terry Allen Miles," Banks said at a New Year's Day news conference. "Our whole focus is to get these girls home safe."
A statewide Amber Alert issued for the girls was discontinued Wednesday by authorities, who said they had evidence that Miles and the children were no longer in Texas.
Lilianais is described as about 5 feet tall and 100 pounds, according to the National Center for Missing and Exploited Children. She has light brown hair and brown eyes, and has braces and a nose stud on the right side of her nose.
Luluvioletta also has brown hair and eyes. She is about 4 feet 5 inches tall and weighs 75 pounds.
Miles is described as 6 feet 2 inches tall and 200 pounds. He has brown hair and hazel eyes, wears glasses and has a beard, the center said.
He and the girls may be traveling in a gray, 2017 Hyundai Accent hatchback with Texas tag number JGH9845. The vehicle has a white sticker on the upper right side of the rear window.
Authorities have not determined a motive for Bates' death or the alleged kidnapping, NBC News reported.
Bates worked as a patient care technician at St. David's Round Rock Medical Center, where administrators said the staff is grieving their loss.
"As a patient care technician at the hospital, Tonya spent her days caring for patients, and we will carry on her legacy by doing what she loved -- caring for our community," Jeremy Barclay, CEO of the hospital, said in a statement obtained by the American-Statesman. "We are keeping Tonya's daughter in our thoughts and prayers, along with the law enforcement officers who are working diligently to bring them home safely."
KHOU in Houston reported that a possible sighting of Miles and the girls was reported in New Mexico. There was also a possibility that the trio was in southern Colorado.
The Austin American-Statesman reported that Miles also has connections to Louisiana, which include multiple domestic violence arrests in Calcasieu Parish, located in the southwestern portion of the state. He was charged with murder in the 2014 death of Pamela Parker in Sulphur, but a grand jury ultimately failed to indict him for the crime.
KPLC in Lake Charles reported that Miles remains a person of interest in Parker's death, as well as in a 2015 rape case. He was also charged with attempted murder in 2011 after he was accused of beating his girlfriend.
"He tried to strangle her and staged a potential suicide by trying to hang her with (a) sheet," Sulphur Police Chief Lewis Coats told the news station.
The charges in that case were dropped because the witness was unable to testify against Miles, Coats said.
The police chief said law enforcement there have been dealing with Miles since 2006, when he moved to Sulphur from California to live with his mother. At that time, Miles was on parole in California.
"He's a man well-known amongst law enforcement for multiple cases of domestic abuse and drugs," Calcasieu Parish Sheriff Tony Mancuso told KPLC.
Authorities did not disclose Miles' connection to Bates or her family, but Bates' Facebook page indicated that she evacuated to the Lake Charles area when Hurricane Harvey hit Texas in August.
It was not immediately clear when Miles moved to Round Rock.
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