It would be the first proposal of its kind.
"I don't think it'll make a difference," said Julie Arnold.
Council member Mike Martinez and two others are considering bringing a vote later this month to take steps to restrict gun shows on private property within the city limits and possibly ban any events on city and county-owned property.
Krista Schumacher said, "I think that's their role and that's why we elect them, you know, to represent us and work for the common good. I think we should definitely restrict guns."
It's a controversial topic that's still too painful for some to discuss.
Less than a month ago, in Newtown Connecticut, a gunman killed 26 students and staff members at Sandy Hook Elementary School.
"It's not gonna stop anytime soon, I don't see it," added Arnold.
The proposal would impact several popular gun shows usually held at the Travis County Expo Center.
Those in support hope banning gun shows in Travis County will lower the distribution and sales of guns in the area.
"I think that they should absolutely restrict them in Texas. I mean, people from all over the world come to the United States to get guns, it's way too easy to get guns here," Schumacher said.
Others, say a gun show ban would do more harm than good to the local economy.
"It can bring in so many tourists. It can bring in tons of money for the county that the gun show's in," said Michael Bierman.
"I am not smart enough to know what the solution is and I think that there is no quick fix," said Erik Torstensson said.
It's something everyone can agree on, a solution to the gun problems we're facing will take time and cooperation from both sides.
The ban would not apply to private gun retailers.
Some other council members have shown support for this.
Kathie Tovo, while she says her office is looking further into the proposal, she's pleased to see the idea come up.