Congress showed a newfound appetite for diving into the gun debate Monday, with key Republicans saying they’re working on bills they think can win bipartisan support and make a dent in what all sides now see as an epidemic of violence.

Senate Majority Leader Mitch McConnell, who for two days was the focus of criticism for lack of action, announced that he has directed key committee chairmen to come up with solutions, as long as they don’t infringe on the Second Amendment.

“Senate Republicans are prepared to do our part,” the Kentucky Republican vowed.

Sen. Lindsey Graham, South Carolina Republican and chairman of the Judiciary Committee, said he’s already struck a deal with Democratic Sen. Richard Blumenthal of Connecticut to write a bill encouraging states to adopt red flag laws, which would allow families or authorities to tip police off to individuals who may pose a danger, and if a quick probe finds cause for concern, to confiscate firearms from those people.

The idea got a boost from President Trump at the White House.

Sen. Lamar Alexander, Tennessee Republican and chairman of the health committee, said he’s interested in seeing what can be done on background checks — an area where Republicans have, in the past, been deeply divided.

Sen. Patrick Toomey, Pennsylvania Republican, announced he’ll try to revive the bill he wrote with Democratic Sen. Joe Manchin of West Virginia to expand background checks to include nearly all firearms transactions. That legislation was defeated by a GOP-led filibuster in 2013, but Mr. Toomey said last weekend’s shooting sprees in Texas and Ohio should change things.

“I don’t know exactly whether we will get a different outcome this time,” Mr. Toomey told reporters on Monday. “I hope that if nothing else the accumulative pain of so many of these horrific experiences will be motivation to do something.”

Democrats said Congress should get started immediately.

House Speaker Nancy Pelosi and Senate Minority Leader Charles E. Schumer called on Mr. McConnell to cancel senators’ summer vacation, which began this week and is planned to run four more weeks.

“It is incumbent upon the Senate to come back into session to pass this legislation immediately,” the two Democratic leaders said.

Mrs. Pelosi said she’s ready to bring the House back should the Senate pass its own solutions. Or, Democrats said, the Senate could pass the background check bill that already cleared the House in February.

That legislation would require gun sales between private parties to undergo a background check. Current federal law only requires sales by a licensed dealer to be screened.

Many Republicans have resisted that, arguing it would be cumbersome, and could make it difficult to gift a weapon to a family member or friend.

The Democratic leaders chided Mr. Trump, who had suggested Monday morning on Twitter that he would call for stiffened background checks but, when he delivered a speech later in the day, didn’t include that demand in his calls for action.

Instead, Mr. Trump said the country needs a culture change that includes trying to reduce the levels of violence in video games, asking social media companies to do more to spot danger cases, and pulling firearms from the hands of the mentally ill.

Mr. McConnell said he would take cues on Senate action from Mr. Trump’s to-do list.

“Only serious, bipartisan, bicameral efforts will enable us to continue this important work and produce further legislation that can pass the Senate, pass the House, and earn the president’s signature,” he said.

Mr. McConnell, who is at home in Louisville recuperating from a broken shoulder, did not address Democrats’ demand that the Senate come back into session.

But Mr. Toomey said that might be “counterproductive.” He said forcing a speedy vote could lead to a partisan showdown, short-circuiting the work required for a thoughtful bipartisan solution.

Universal background checks have been a Democratic priority for years, well before this weekend’s shootings — though it’s not clear whether they would have prevented either attacker from getting his weapon.

The police chief in Dayton, where one of this weekend’s shootings took place, said the suspect in that rampage bought his .223 rifle from a local dealer.

Connor Betts, who killed nine people in 30 seconds before being slain by police, didn’t have anything in his background that would have triggered a background check flag preventing the purchase.

Mr. Toomey, speaking to reporters, acknowledged his background check bill wouldn’t stop every instance of mass shootings, but he said it could diminish the risks. “I hope that we’re at a moment where the appetite has changed,” he said.

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