Tuesday's inaugural session focused on school districts arming teachers and better tracking student mental health.
Abbott, a Republican, has vowed that any changes to state laws would "protect Second Amendment rights".
The last day of school in the Arlington ISD is June 1.
Abbott spokesman Stephen Chang said the governor was finalizing plans for the roundtables, including the dates and participants.
A teenage boy charged with fatally shooting eight students and two teachers during a gun rampage at a Houston-area high school had been spurned by one of his victims after making aggressive advances, her mother told a newspaper.
Although Abbott asked the country for support, for both the families impacted and first responders, his calls for action supersede prayers. Police said Pagourtzis confessed to Friday's killings after he was taken into custody, but authorities have offered no motive yet for the massacre, the fourth-deadliest mass shooting at a USA public school in modern history.
A church where one of the victims of a Texas school shooting attended services is providing a licensed counselor during its worship to offer advice on how to deal with tragedy.
Safety options such as providing fewer entrances to school buildings or having children arrive at earlier times were proposed by Patrick on ABC's "This Week".
“She was a great soul,” Aziz Sheikh said Saturday of his daughter Sabika. He shows her
Had it not been for God, Abbott said he too would still be stuck. We have to look at ourselves, George. "Can there be gun regulation, gun control?"
"If the death penalty were on the table within the punishment range, I believe that we would be seeking it in this case", Roady said.
Pagourtzis used his father's legally owned shotgun and.38-revolver in the massacre, officials said.
In the footage, obtained by TMZ, Dimitrios Pagourtzis is seen wearing traditional Greek clothing while dancing with other members of a Greek Orthodox Church days before he allegedly killed 10 people and wounded 13 more in an attack at Santa Fe High School.
Another school banned backpacks to prevent hidden weapons from slipping through, while others changed dress codes to forbid heavy clothing such as the trench coat Pagourtzis was said to be wearing to hide his weapons.
"And I can assure you that whatever the answers we find from the experts, and from citizens and from parents, and all those involved, that will be Senate Bill One", Patrick said. It could have been the school of a member of your family. The attacker wore a black trench coat to school even in the Texas heat on Friday. We've had enough moments of silence. "I think that she knew beforehand because this guy kept harassing her about wanting to be with her, like wanting to be in a relationship", Baze told KENS5. Personally, I grew up in Texas with guns. The first one features officials from districts that arm some teachers or contract with local police for security. We were mighty proud. You can do it, but it's not what you really need. And gunowners have nothing to fear from a quick, universal background check. "It's not about the guns, it's about us".
The third roundtable discussion will take place Thursday, May 24.