Congress passed a similar measure in 2007 after the shooting at Virginia Tech, hoping to improve the reporting of criminal records and mental health information to the gun purchasing database. It did not head off Sutherland Springs, where a gunman legally bought his firearm despite having been convicted of domestic abuse.

Yet at least two, and possibly, three Republican senators are blocking the proposal from quick consideration. And Democrats — whose leader, Senator Chuck Schumer of New York, is a co-sponsor of the measure — do not want to allow it to move forward for a vote unless it is paired with more far-reaching legislation, like an expansion of background checks and a ban on assault weapons.

“He comes to the floor every day and says, let’s do the small Fix NICS bill and then we’ll see about other proposals,” Mr. Schumer said Tuesday, referring to Mr. Cornyn. “He knows, as well as I do, that Fix NICS is not even close to a response to the epidemic of gun violence in the country.”

On the other side of the Capitol, the House is expected to vote Wednesday on a measure, introduced in response to the Parkland shooting, to beef up safety at the nation’s schools.

That bill has nothing to do with guns; it would authorize $50 million annually for safety improvements, including training teachers and students in how to prevent violence and developing anonymous reporting systems for threats. A companion measure in the Senate would also give schools money for physical improvements, such as installing metal detectors or bulletproof windows.

“I’m extremely interested in seeing Senator Cornyn’s Fix NICS bill pass, and a significant school safety bill pass,” Mr. McConnell said Tuesday. “The best way to get that done is still under discussion, but I’m anxious to pass both of them and pass both of them soon.”

A former Texas attorney general and justice of the Texas Supreme Court, Mr. Cornyn, 66, grew up around guns. He keeps his license to carry a concealed weapon in his wallet — “My wife and I both got it together,” he said — and recalls learning to shoot as a teenager, with a .22-caliber rifle at target practice.