House Majority Leader Eric Cantor (R-Va.) speaks at the American Enterprise Institute on Feb 5. (Mark Wilson/Getty Images)

The second-ranking House Republican said Tuesday that he supports improving the federal background-check system for gun buyers but stopped short of endorsing universal checks on all weapon purchases.

The comments by House Majority Leader Eric Cantor (R-Va.) came as two GOP lawmakers from suburban districts announced plans to co-sponsor legislation to make gun trafficking a federal crime for the first time. The moves signal potential openings for bipartisan compromise on gun control, a debate so far dominated by Democrats with little said or done by Republicans.

Cantor, giving the most specific comments on gun control by a GOP congressional leader since President Obama outlined his proposals in late January, told CNN in an interview that lawmakers could consider adopting a plan implemented by Virginia after the 2007 shootings at Virginia Tech. Since the shootings, the state has linked mental-health information to law enforcement databases used to conduct background checks for gun purchases.

“I think that we can take a lot of lessons from what Virginia did and put it in place at the federal level,” Cantor told CNN. “Because there’s a lot of states that aren’t doing what Virginia is doing to try to beef up the database for the background checks to make sure that we actually can do something that does have a chance at reducing the likelihood and hopefully eliminating that from happening again.”

When asked, Cantor stopped short of saying he supported universal background checks, saying only that “I am for making sure that we increase the quality of information in the database that is in existence already.”

President Obama proposed expansive gun-control policies aimed at curbing gun violence. The Obama administration can implement about half of the proposals, but the others — arguably some of the more critical initiatives — will require congressional approval.

Cantor aides declined multiple requests Tuesday to clarify the leader’s comments.

Last month, Obama took four separate executive actions designed to improve the information in the existing background-check system. The administration has begun removing legal and regulatory barriers for states to share information about the mentally ill with the federal background-checks database and is holding federal agencies accountable for sharing relevant information with the database. The administration is also asking Congress to provide $50 million in incentives for states to share criminal history and mental-health information with the federal database.

On Capitol Hill, Rep. Elijah E. Cummings (D-Md.) said Cantor’s comments “clearly opened the door for the House to move on meaningful legislation” to address gun violence.

Cummings and Reps. Carolyn B. Maloney (D-N.Y.), Patrick Meehan (R-Pa.) and Scott Rigell (R-Va.) on Tuesday unveiled the first bipartisan gun-control bill introduced in the House since the Dec. 14 shooting at Sandy Hook Elementary School in Newtown, Conn.

The bill would make gun trafficking a federal crime and impose new penalties against gun “straw purchasers” who knowingly buy firearms for convicted criminals who are barred from buying their own weapons. It mirrors a bipartisan Senate bill introduced last week and was part of Obama’s recent proposals.

Rigell and Meehan said they have discussed the bill with Republican colleagues. “There are some questions, but generally they’re very supportive,” Rigell said of potential co-sponsors.

Cummings recalled the shooting death last year of his 20-year-old nephew: “It is a painful thing to see your blood splattered on the walls of an apartment. To see tissue from your loved one splattered on walls.”

Also Tuesday, the White House announced that Obama will honor the six Sandy Hook Elementary School teachers and administrators who died trying to protect their students during the Newtown massacre. They will posthumously receive the 2012 Presidential Citizens Medal, one of the highest honors bestowed by the president, at a Feb. 15 ceremony, a White House official said.