GRAPEVINE, Texas (CBSDFW.COM) – Following the deadly shootings in El Paso and Dayton, Ohio, President Donald Trump and some Republicans, including Texas Senator John Cornyn, say they’re willing to consider red flag laws to keep guns away from people who shouldn’t have them.

Some conservatives say they’re not convinced, something Sen. Cornyn acknowledged. “There is a concern about due process and about abuses, but we’re interested trying to do whatever we can to save more lives.”

In Grapevine Thursday, Senator Cornyn said lawmakers need to also consider making it easier for professionals to share information about potentially dangerous people before they act.

Sen. John Cornyn in Grapevine (CBS 11)

“The biggest challenge we have is we know there are ticking time bombs out there, but until somebody has committed an offense and had an encounter with law enforcement, there’s no way to identify them unless you get information from teachers, counselors, parents, neighbors, and the like and there’s a lot of obstacles about sharing that information that we need to look at,” he said.

He also said lawmakers should consider updating federal laws against domestic terrorism.

Cornyn met with Senator Ted Cruz, Governor Greg Abbott and other state leaders in El Paso Wednesday to discuss what more can be done.

During a news conference Wednesday, Governor Abbott said he’s open to any idea.

Cornyn said they discussed at least one idea.

“I think the Governor is interested in what Texas can do to improve the compliance with the background check system. What that means of course is people are already legally disqualified from purchasing firearms to make sure they can’t lie and buy because the background check system doesn’t have their name as it should,” Sen. Cornyn said.

The Governor announced Wednesday that he will hold a series of roundtable discussions with experts to consider ways to prevent mass shootings in Texas.

The roundtables would be similar to those he held last year following the deadly Santa Fe High School shooting.

This year, the legislature passed laws to improve school safety and mental health but not red flag laws.

One Democratic State Representative, Victoria Neave of Dallas, said she wants the Governor to call a special session instead of holding the roundtable discussions.

On Capitol Hill, Democrats, including Congressman Marc Veasey of Fort Worth, have pressed Senate Republicans to return from recess to act on their bills expanding background checks.

“We’ve passed a background check bill. It’s sitting on Mitch McConnell’s desk right now,” said Rep. Veasey.

Sen. McConnell, the Senate Majority Leader, has said he asked some of his committee chairmen to come up with good ideas Senators can consider when they return after Labor Day.

Cornyn said it’s important to find out more about the El Paso and Dayton shooters first.

“I don’t think as much as I would love to get it done today, I think there’s going to be a lot of information that’s still going to be coming in from Dayton and from the investigations in El Paso that would help inform the decisions we make,” said Cornyn.

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