WASHINGTON — In a phone call with President Trump about gun safety legislation last week, Senator Joe Manchin III made an unusual offer: If Mr. Trump signed onto a background check bill that he and Senator Pat Toomey have been pushing for six years, the president could claim the naming rights for himself.
“This won’t be Manchin-Toomey,” Mr. Manchin recalled telling the president, referring to the measure that would expand background checks for gun buyers. “This is going to be the Common Sense Gun Bill of 2019 by President Trump.”
With Mr. Trump expected to decide as early as next week whether he will defy the National Rifle Association and embrace a bill to expand background checks for gun purchases, Mr. Manchin, Democrat of West Virginia, is hardly the only lawmaker appealing to the image-conscious president. In meetings and phone calls with Mr. Trump and his advisers, backers of various gun safety bills have been waging a full-scale Washington courtship rooted in policy, politics and a little bit of presidential ego.
Speaker Nancy Pelosi and Senator Chuck Schumer, the top two Democrats in Congress, tried to tap into Mr. Trump’s penchant for drama, promising to join the president for a “historic signing ceremony at the Rose Garden” if he abandoned his veto threat and got behind a more expansive, House-passed background checks bill.
Mr. Toomey, Republican of Pennsylvania, has appealed to “inherent logic,” he said, making the case “on the merits” of using backgrounds checks to keep guns from criminals during a string of conversations with the president, most recently at the White House Thursday.
And Senator Chris Murphy, Democrat of Connecticut and a central player in the background checks negotiations along with Mr. Manchin and Mr. Toomey, has told the president that with 90 percent of the public favoring expanded background checks, there would be “enormous benefit to him if he was willing to be the guy who brought this to completion.”
When Mr. Manchin volunteered to take his name off the background checks bill, Mr. Murphy said, the three senators were huddled around a speakerphone, talking to Mr. Trump from Mr. Toomey’s Capitol hideaway office.
“The president is a guy who thinks about branding,” Mr. Toomey said. “I think we just want to make it clear to the president that if he was going to do this, that he should get credit for it.”
Gun safety has long been one of the most intractable issues in Washington, but it catapulted to the forefront of the congressional agenda after a series of mass shootings over the summer. Democrats immediately demanded the Republican-controlled Senate pass a bill that had already moved through the House and would require all gun buyers undergo background checks. But that bill remains stuck in the Senate, where Senator Mitch McConnell, the Republican leader, refuses to take it up, citing Mr. Trump’s veto threat.
The somewhat narrower Manchin-Toomey bill would extend background checks to all commercial sales, including those at gun shows and over the internet. But that bill has fallen twice to a Senate filibuster, once in 2013 and once in 2015. Only two Republicans who have previously voted in favor of it — Mr. Toomey and Senator Susan Collins of Maine — remain in the Senate today.
Despite those obstacles, the political landscape around gun safety does appear to be changing. The N.R.A. is in a weakened state, mired in internal conflict and facing investigations into possible abuse of its tax-exempt status. The gun safety movement is ascendant, fueled by student activism in the wake of last year’s mass shooting at Marjory Stoneman Douglas High School in Parkland, Fla.
And several newly elected Republicans, including Senators Mitt Romney of Utah and Mike Braun of Indiana, have called for expanded background checks in the wake of the summer shootings. Senator Lindsey Graham, Republican of South Carolina and a close ally of the president, is supporting a different type of gun safety legislation known as a “red flag” bill, and has warned Mr. Trump it was untenable to tell the public there was nothing he could do, “because the shootings are going to keep coming.”
Ms. Collins, who said she has been in frequent contact with the president’s advisers though not Mr. Trump himself, said he is convinced that “there are a lot of responsible, common sense gun safety provisions that could be bundled together that we could pass.”
But both Republicans and Democrats agree that nothing will pass without Mr. Trump’s support. The president is said to be reviewing a package of ideas presented to him by White House advisers. But he is keeping his views to himself, and his earlier public statements have been all over the map, in keeping with his history of flip-flops on the gun issue.
At first, after back-to-back massacres in El Paso and Dayton in early August, Mr. Trump said there was “tremendous” support for “really common-sense sensible, important background checks” and bragged that he could overcome the longstanding Washington gridlock on the issue.
“There’s never been a president like President Trump,” he said then.
“At that point, he was positively over the moon about background checks,” said Mr. Murphy, who sponsors the Senate version of the House background checks bill.
But more recently, the president has walked back those enthusiastic comments, while critics of the Manchin-Toomey bill have raised two primary objections. They say the current system, which relies on federally-licensed firearms dealers conducting background checks, imposes a burden on rural buyers who live far from such dealers. And they warn that background check records could be used to create a national firearms registry, which they deeply oppose.
Earlier this week, William P. Barr, the attorney general, circulated a proposal on Capitol Hill designed to address those objections. The plan would create a new class of “licensed transfer agents” who would not sell guns, but would be authorized by the government to conduct background checks. And sellers, not firearms dealers or transfer agents, would retain the records, so they could not be used to create a registry.
In interviews, Senators Graham, Manchin, Toomey and Murphy all said Mr. Barr seemed serious, and that the plan formed the foundation of a potentially workable compromise.
“As long as it enables him to do his job and the Department of Justice to do it’s job, I’m fine,” Mr. Manchin said.
“I’ve told him what Barr is working on seems to be sensible to me,” Mr. Graham said, recounting his most recent conversation with the president on the issue. “I don’t know exactly what the final product is. I will support whatever the president supports.”
But the White House took pains to distance itself from Mr. Barr’s proposal, and emphasized that it does not have Mr. Trump’s blessing.
And any compromise in the Senate will be difficult to achieve. Mr. Toomey will have to bring in his colleagues from the right, who are leery of anything that smacks of gun control, and Mr. Murphy will have to bring in his colleagues from the left, who are demanding Senate passage of the stricter House bill and do not want to hand the president any political victories just as Mr. Trump is seeking re-election.
“I said, ‘Mr. President, I’m as hard-line as you come on guns, and so my hope is that if I signed onto a compromise, that other people may be willing to take a hard look at this,” Mr. Murphy said.
In the meantime, proponents are hoping to appeal to Mr. Trump’s oft-stated desire for a big deal.
“My gut tells me that the president wants to do something meaningful to do more to keep guns out of the hands of people who should not have them,” Mr. Toomey said.