McConnell said Thursday that he had met with Cornyn, who flew back to Texas this week after a shooting at Robb Elementary School in Uvalde, west of San Antonio, left 19 children and two teachers dead.
McConnell said in an interview on CNN that he had urged Cornyn to talk to Sens. Kyrsten Sinema of Arizona and Chris Murphy of Connecticut, two Democrats who have called for bipartisan talks on gun safety legislation, "and others who are interested in trying to get an outcome that’s directly related to the problem."
"And so I am hopeful that we could come up with a bipartisan solution that’s directly related to the facts of this awful massacre," the Kentucky Republican said.
Asked if the legislative solution that Republicans would welcome involves red flag laws, background checks or mental health access, McConnell declined to offer any specifics. He only said that the goal is to "come up with a proposal, if possible, that’s crafted to meet this particular problem."
He also said he plans to keep in touch with Democrats and hopes that Congress can reach a deal that can "actually pass and become law rather than just scoring points back and forth."
Democrats have warned that the issue of gun violence will be on the ballot in the midterm elections this fall. Polls in recent years have consistently indicated overwhelming support for universal background checks.
Earlier in the day, Murphy, a leading proponent of passing stricter gun control measures, said that members from both parties planned to have a discussion on the nation's gun laws Thursday, before leaving for Memorial Day recess.
The group, which included Sinema as well as Democratic Sens. Joe Manchin of West Virginia, Richard Blumenthal of Connecticut and Martin Heinrich of New Mexico, met in Murphy's "hideaway," a small private space he has in the basement of the Senate.
Sens. Susan Collins of Maine, Lindsey Graham of South Carolina, and Pat Toomey of Pennsylvania were among the Republicans who joined the meeting. Sen. Bill Cassidy, a Louisiana Republican, also attended by phone.
They will break off into smaller working groups to focus on specific pieces of a potential package, negotiators said. It's unclear what the new bipartisan group will be able to accomplish, if anything.
The Democratic-controlled House has passed legislation that would expand background checks and tighten gun regulations. Republicans, however, have stymied consideration of those bills in the Senate.
The chamber came close to passing gun restrictions in 2013 — several months after the Sandy Hook massacre — when lawmakers considered an amendment, proposed by Manchin and Toomey, to expand background checks on all commercial gun sales. But the Senate rejected the measure in a 54-46 vote.
Any proposal addressing guns would require 60 votes to advance to a final vote in the upper chamber, which means Democrats would need at least 10 Republicans to join them in supporting a measure for it to pass.