Sen. John Cornyn (R-Texas) said on Tuesday that he will introduce legislation aimed at strengthening the gun background check system in the wake of a deadly mass shooting in his home state.

"I plan to introduce legislation ... to ensure that all federal departments and agencies, including the Department of Defense, upload the required conviction records into the national database," Cornyn said.

The Texas Republican, citing the Justice Department, added that the number of records currently being shared with the National Instant Criminal Background Check System (NICS) is "staggeringly low."

"That is unacceptable and it must change."

Devin Kelley, the identified gunman in the massacre at First Baptist Church in Sutherland Springs, Texas, received a "bad conduct" discharge from the Air Force in 2014 after being court-martialed on a domestic violence charge.

Kelley's court-martial conviction should have been reported to the FBI's database. Had it been, it may have been harder for him to purchase a firearm legally.

But Air Force officials on Monday said the Holloman Air Force Base Office of Special Investigation did not enter Kelley's information into the system.

Lawmakers have homed in on the military's reporting system after the church shooting.

Democrats, meanwhile, have used the shooting to urge Congress to pass tougher gun and background check laws, but those bills likely face an uphill battle in a GOP-controlled Congress.

Cornyn, asked about legislation restricting guns, told reporters on Tuesday that "I think that gets us into a stalemate again."

"I'm talking about making sure that current law is enforced. ... It may require additional resources," he said.