Meanwhile, congressional Republicans — who are wary of any new gun restrictions but have been under public pressure for weeks to do something in response to the tragedies — are going on the ­offensive, highlighting comments from former congressman Beto O’Rourke of Texas at Thursday’s Democratic presidential debate when he vowed to seize assault weapons from gun owners through a mandatory buyback program.

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At the House Republicans’ annual retreat in Baltimore on Friday, House Conference Chairwoman Liz Cheney (Wyo.) said talk of gun confiscation will “absolutely” motivate people to vote Republican.

“I think it’s absolutely clear that there’s one party here that absolutely will make sure that we are defending the Second Amendment, defending Americans’ constitutional rights,” Cheney said. “There is another party — and you’ve seen it with the Democrats — who really have no care and no concern about the Second Amendment.”

The primary focus remains on the White House, where weeks have passed since a pair of mass shootings in El Paso and Dayton, Ohio, galvanized public pressure for new gun measures that have since been prioritized by congressional Democrats.

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The White House is looking into a number of ideas, including a version of expanded background checks and additional resources for addressing mental-health issues. Attorney General William P. Barr has been drafting legislation that would expedite the death penalty for people accused of mass shootings — although questions have been raised about how much of a deterrence that would be given that the perpetrators of mass shootings often expect to be killed in the massacre.

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But even people close to the White House, including those who have spoken to the president frequently on potential gun measures, remain unsure of what Trump will embrace.

“I’m sort of waiting for the baby to be born here,” Sen. Lindsey O. Graham (R-S.C.), who is working on his own modest gun restrictions, said Friday.

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Shortly after he was briefed by advisers on potential policy options Thursday, Trump told reporters at the White House that a “lot of progress was made” but gave no specifics.

Whether he would support expanded background checks was up to Democrats, he said.

“If this is a movement by the Democrats to take your guns away, then it’s never going to happen because we’re never going to let that happen,” Trump said. “We will always be there for our Second Amendment.”

Privately, the White House has floated at least one particularly unorthodox idea: an app connected to the National Instant Criminal Background Checks system that could be used to conduct background checks on private gun sales, according to three senators and other officials familiar with the proposal who spoke on the condition of anonymity because the proposals have not been finalized.

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In turn, lawmakers and congressional aides have pointed out to the White House potential flaws in the proposal, such as security issues with the app and how it would affect how law enforcement officials access records on gun ownership.

Senate Majority Whip John Thune (R-S.D.), who said he had been told of the idea by one White House official, raised concerns about gun owners’ personal data being so freely available on a handheld device, and said it “would kind of be a de facto registry.”

“It’s fraught with a lot of issues when it comes to some of the Second Amendment concerns,” Thune said.

Trump himself has sent confusing signals in his conversations with senators.

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When the president spoke by phone earlier this week with Sens. Joe Manchin III (D-W.Va.), Patrick J. Toomey (R-Pa.) and Chris Murphy (D-Conn.) — three critical lawmakers in the background-check debate — he appeared to know little about what his administration was proposing or what he was going to do about guns, according to people familiar with the discussion who spoke on the condition of anonymity to describe the private conversation. Trump asked questions about various proposals that have been in the news for weeks.

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He veered from asking senators about vaping to complaining about CNN and bragging about GOP wins in a pair of closely watched special elections in North Carolina, according to these people. The senators gave him plaudits on vaping but were silent during his boasts about his political victories.

Trump met with senior staff Thursday afternoon, including the domestic policy council, aides to Vice President Pence and other political and policy officials.

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Trump’s aides have given him data that show his supporters are skeptical of expanding background checks and there are many high-intensity voters who would “really oppose it,” according to one political adviser who spoke on the condition of anonymity to discuss the private polling. A number of Trump’s aides have argued against any expansive background-check bill.

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Two officials involved in the negotiations said many players in Trump’s orbit are amenable to some background-check law. But Michael Williams, a White House official who once worked for the National Rifle Association and is close to acting chief of staff Mick Mulvaney, has opposed an expansive background checks agreement like some Democrats want and has seemed to slow any progress, both officials said.

Yet in the Senate, proponents of expanding background checks have worked to influence Republicans who either weren’t in the Senate for the 2013 debate on guns or have indicated that they are rethinking their opposition to the legislation drafted by Manchin and Toomey.

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One is Graham, a close Trump ally who chairs the committee that oversees gun policy. He said Thursday that he is open to supporting the 2013 background-checks bill, as long as he secures changes that he is discussing with its chief authors.

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Graham indicated that he is rethinking the issue after attacks in Odessa, Tex., in which the gunman had not undergone a background check when he obtained the rifle used in the Aug. 31 massacre. The shooter, Seth Ator, had previously failed a background check.

“The Odessa case is a good example,” Graham said. “Guy was mentally ill. Could not get a gun when he went to the gun stores because he was in the background check system. Bought a gun from a nonfamily, non-friend person, who’s selling guns for money. So I’d like to capture that transaction.”

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Democrats have expressed varying levels of willingness to negotiate with the White House — particularly on the expansion of background checks.

Party leaders in both chambers have urged Senate Republicans to take up a House-passed bill that would require background checks for commercial sales over the Internet and at gun shows and would also require private transfers to be screened through a federally licensed gun dealer.

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The White House has threatened to veto that bill, and some Senate Democrats have discussed a narrower expansion of background checks along the lines of the Manchin-Toomey proposal, which would require them for all commercial sales while leaving private sales untouched.

Democrats will likely have to wrestle with how much they are willing to retreat from their zero-loophole stance. Many Republicans, for instance, are dead set against restricting private sales or transfers — frequently raising the example of a parent giving a treasured rifle to a child or a person lending a handgun to a trusted friend threatened by an abusive partner.

“If it’s short of universal background checks, it doesn’t solve the problem,” said Rep. Mike Thompson (D-Calif.), who is chairman of the House Gun Violence Prevention Task Force and has had conversations with White House officials. “I want to see what they have to offer. But the fact of the matter is there’s some things that they could do that just won’t cut the mustard.”