WASHINGTON, D.C — A cornerstone of President Barack Obama’s drive to check gun violence is gathering bipartisan steam as four senators, including two of the National Rifle Association’s congressional champions, privately seek compromise on requiring far more firearms purchasers to undergo background checks.

The talks are being held even as Obama’s call to ban assault weapons and high-capacity ammunition magazines, the two other major pillars of his plan, are hitting rough waters on Capitol Hill. An agreement among the four senators to expand background checks would add significant impetus to that high-profile proposal by getting the endorsement of a group that ranges from one of the Senate’s most liberal Democrats to one of its most conservative Republicans.

“We’ll get something, I hope. I’m praying for it,” said Sen. Joe Manchin, D-W.Va., one of the participants.

Manchin, a moderate Democrat, is an NRA member who aired a 2010 campaign ad in which he literally shot a hole through Democratic environmental legislation that he pledged to oppose.

Also involved is Sen. Tom Coburn, R-Okla., another NRA member with a strong conservative record but occasional maverick impulses; No. 3 Senate Democratic leader Charles Schumer of New York, a liberal; and moderate GOP Sen. Mark Kirk of Illinois.

Background checks are required only for sales by the nation’s 55,000 federally licensed gun dealers, but not for private purchases like those at gun shows, online or in person. There are few indisputable, up-to-date statistics on how many guns change hands without background checks, but a respected study using 1990s data estimated that 30 percent to 40 percent of gun transactions fit into that category.

The senators’ talks have included discussions about how to encourage states to make more mental health data available to the federal system for checking gun buyers’ records, according to people who spoke anonymously because they were not authorized to describe the private negotiations. They are also considering potential exemptions to expanded background check requirements, including transactions involving relatives or people with licenses to carry concealed weapons

People involved in the talks would share little about their substance. In one of the few public remarks about the talks by participants, Schumer said last week that the talks have been productive and said the package they were seeking “will not limit your ability to borrow your Uncle Willie’s hunting rifle or share a gun with your friend at a shooting range.”

Congress has been focusing on guns since the December massacre of 20 first-graders and six adults at a school in Newtown, Conn. Senate Judiciary Committee Chairman Patrick Leahy, D-Vt., wants his panel to approve gun control legislation in the next few weeks and has voiced strong support for universal background checks for firearms purchases.

While an expansion of background checks is expected to be a key part of any gun control bill Leahy produces, a version of that provision with bipartisan support could give the entire package a boost.

It is likely that any gun-control bill will need 60 votes to pass the 100-member Senate. Democrats have 55 votes, including two Democratic-leaning independents.

Leaders of the GOP-run House are planning to see what, if anything, the Senate passes before moving on gun legislation. Strategists believe that a measure that passes the Senate with clear bipartisan support could pressure the House to act.

The political impact that the four senators could have by reaching agreement stems largely from who they are.

If Coburn embraces an agreement, that could help win over other conservative Republicans at a time when the GOP is responding to its White House and congressional election losses of last November by trying to broaden its national appeal.

In an Associated Press-GfK Poll last month, requiring more background checks got overwhelming public support, compared to just over half who backed bans on assault weapons and high-capacity ammunition magazines.

“The whole goal is to keep guns out of the hands of the mentally ill and criminals,” Coburn said in a brief interview.

Manchin’s support could make it easier to win backing from other Democratic senators from GOP-leaning states, many of whom face re-election next year and who have been leery of embracing Obama’s proposals.

“If the language is meaningful, it would be obviously a huge step,” said Josh Horwitz, executive director of the Coalition to Stop Gun Violence, which represents child welfare, religious and other groups favoring gun curbs. “To have someone like Coburn, who’s voted consistently with the gun lobby, to come out and endorse a meaningful background check would be very helpful.”

Schumer and Kirk each have “F” scores from the NRA for their voting records in Congress, while Coburn and Manchin have “A” ratings.

Though widened background checks is given the strongest chance for enactment of Obama’s major proposals, it is opposed by the NRA and many congressional Republicans, who consider it intrusive and unworkable for a system they say already has flaws.

“My problem with background checks is you’re never going to get criminals to go through background checks,” Wayne LaPierre, NRA executive vice president, told the Senate Judiciary Committee at its gun control hearing last week.

“That’s the way reductions in liberty occur, when you start saying people have to sign up for something and they have a database where they know exactly who’s who,” Sen. Orrin Hatch, R-Utah, said in an interview.

Gun control supporters note that federal laws specifically forbid the national background check system from being used as a registry of gun owners. Much of the information the system collects must be destroyed within a day.

NRA spokesman Andrew Arulanandam declined to comment on the senators’ discussions.

According to Justice Department estimates, the federal and state governments ran 108 million background checks of firearms sales between 1994 when the requirement became law and 2009. Of those, 1.9 million — almost 2 percent — were denied, usually because would-be purchasers had criminal records.

People legally judged to be “mentally defective” are among those blocked by federal law from firearms purchases. States are supposed to make mental health records available to the federal background check system and receive more generous Justice Department grants if they do, but many provide little or no such data because of privacy concerns or antiquated record-keeping systems.

People following the discussions say the talks have touched on:

—The types of family relatives who would be allowed to give guns to each other without a background check.

—Possibly exempting sales in remote areas.

—Whether to help some veterans who sought treatment for traumatic stress disorder — now often barred from getting firearms — become eligible to do so.