After the Senate voted down four gun-control bills inspired by the Orlando massacre, the Maine Republican Susan Collins unveiled compromise legislation Tuesday that addresses a bipartisan concern: individuals suspected of terrorist ties being able to purchase guns.

“All of us are united in our desire to getting something significant done on this vital issue,” Collins said at an afternoon press conference, flanked by seven Democratic and Republican senators who worked on the legislation. “Surely the terrorist attacks in San Bernardino and in Orlando that took so many lives are a call for compromise, a plea for bipartisan action.”

In a white paper, Collins’s office described the major provisions of the amendment: It prohibits gun sales to people on two terrorist watch lists, including the No Fly List; it allows for American citizens and green-card holders to appeal if their purchase is restricted and to get attorney fees recouped if they win; and it includes a “look-back provision” that requires FBI notification if someone who’s recently appeared in a broader terrorism database buys a gun. At the press conference, Collins noted that the total number of people on the two restricted lists is roughly 100,000, most of whom are foreign nationals.   

The New Hampshire Republican Kelly Ayotte emphasized that the legislation protects Americans’ due-process rights, a major concern among Republicans who voted against a Democratic bill restricting sales on Monday. That proposal prohibited sales to more people than the Collins bill does. Senate Republicans preferred a measure that would have forced the FBI to get a court order when officials wanted to stop a suspected terrorist from buying a gun. Florida Representative David Jolly, a Republican, introduced legislation in the House Tuesday that would also address due-process worries.

Collins’s proposal comes as congressional lawmakers have faced criticism for their inaction in the wake of mass shootings. Democrats, including President Obama, have pushed those critiques, and they lamented Monday’s failed votes. But all “were designed to fall short and therefore demonstrate a lack of consensus in the Senate on guns,” as Russell Berman reported this week. “Republican leaders subjected each of the four amendments to 60-vote threshold and rallied their members around proposals backed by the NRA.” On Tuesday, senators pushed their fellow members to see value in a bipartisan proposal. The Virginia Democrat Tim Kaine suggested gun victims would be waiting “forever” if Democrats and Republicans were to wait for their ideal bills to be passed.  

Senators seemed to anticipate pushback from gun-rights advocates as they described the legislation’s provisions and limitations. Senator Lindsey Graham addressed the powerful National Rifle Association directly in his remarks, telling members he understands their concerns about gun control infringing on the Second Amendment. “But every right—whether speech or buying a weapon or every other constitutional right—has boundaries on it,” said Graham, who noted he himself owns an AR-15 rifle. He added: “If we can’t pass this, it truly is a broken system up here.”

Collins said Majority Leader Mitch McConnell will allow a vote on the proposal soon. Even if it passes, though, it might not be well received in the House. As Roll Call reported Tuesday, House GOP leadership, including Speaker Paul Ryan, have not shown enthusiasm post-Orlando for gun-control measures.