Alice Tripp, legislative director for the Texas Rifle Association, said that her group's 37,000 members were cautious about any attempt to expand laws that can be used to seize or deny guns.
Meanwhile, Pagourtzis's father told The Wall Street Journal newspaper that he suspected his 17-year-old son had been bullied in school.
Signs reading "Santa Fe Strong" could be seen outside many local churches and businesses over the weekend.
Calls for tighter gun regulations in the United States have been prominent following the Marjory Stoneman Douglas High School shooting in Parkland, Florida on Valentine's Day, where 17 students and teachers were killed.
The talks came in the aftermath of Friday's massacre at Santa Fe High School that killed eight children and two teachers in the latest example of the mass shootings that have become a pervasive part of American life.
Abbott said he hoped some of the ideas could be implemented by the start of the next school year, such as an improved threat-assessment system that would allow schools across Texas to better share information.
Texas has 5.4 million students enrolled in its public schools and any changes it made statewide would be costly. He said last week that he wants to find ways to keep guns away from those who pose an "immediate danger to others".
The shooting rampage in Florida sparked a gun control campaign by students and parents that has piled on pressure nationwide for lawmakers to enact gun control legislation. "It is unchallenged at the state level", Jillson said.
In fact, at a church service Sunday, Santa Fe High student Monica Bracknell, who survived the shooting, told the governor the attack should not be turned into a battle over gun control.
Sentiments like those could give Abbott political cover if his roundtable discussions don't lead to major changes.
"We did more today than talk", the governor said. During a press conference Monday at Santa Fe ISD's administration building, Galveston County Sheriff Henry Trochesset revealed more about the timeline between the moment the shooter opened fire and when and how long officers engaged the suspect. "From what I see, I don't believe any of the individuals that were killed were from law enforcement", Trochesset said.
"Many of these young boys have been on Ritalin since they were in kindergarten", he told Fox News. The problem is innocent people are being shot.
Almost 200 Texas school districts allow staff to carry guns.
Still, Lizzie Pannill Fletcher, a lawyer who is favored to win a runoff for the Democratic nomination in a potentially flippable, Republican-held congressional district in Houston, not far from where Friday's shooting took place, refrained from criticizing the governor for not doing more than organizing discussions.