Democrats’ top leaders asked Americans to demand action on gun control and said the Senate must cancel its summer vacation and return to Washington to pass legislation “immediately,” after a pair of shooting rampages this weekend.

House Speaker Nancy Pelosi and Senate Minority Leader Charles E. Schumer said the House, controlled by Democrats, already “did its duty” by passing a bill to expand background checks to cover sales between private parties. Currently federal law only requires sales by a licensed dealer to be screened.

But that legislation, which cleared the House on a near-party line vote in February, has not seen any action in the GOP-controlled Senate, which began a five-week vacation this week. The House is also out on a six-week vacation.

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

Sen. Patrick Toomey, a Pennsylvania Republican who’s working on a bipartisan background check bill in the Senate, cautioned against a rush to bring Congress back. He said forcing a vote now might be “counterproductive,” while letting lawmakers negotiate could pave the way to passage.

Mr. Toomey has for years been working on a bill with Sen. Joe Manchin, West Virginia Democrat, to expand the number of gun purchases subject to a background check.

When their bill was brought to the floor for a vote in 2013, when the Senate was under Democrats’ control, it garnered 54 votes — but that was shy of the 60 needed to overcome a filibuster.

“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.”

He said he would also like to see action on so-called “red flag” legislation that would encourage states and localities to have a process to temporarily take guns from the hands of dangerous individuals. And he called for passage of so-called “lie-and-try” legislation that would report people who, while ineligible to buy a firearm, try anyway. In many cases the attempt itself is a felony, though it’s rarely prosecuted, he said.

The Democratic leaders, in their statement Monday, said if Washington is unable to act it will be proof that the pro-gun lobby is too powerful.

They chided President Trump, who on Twitter Monday morning had suggested he would call for stiffened background checks as part of his solutions to the gun violence.

But when he spoke, Mr. Trump did not mention background checks, instead calling for 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.

Mrs. Pelosi and Mr. Schumer said Mr. Trump, in backing away from his suggestion of stiffer background checks, showed he “remains prisoner to the gun lobby and the NRA.”

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 the attacks.

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. He didn’t have anything in his background that would have triggered a background check flag preventing the purchase.

Mr. Toomey, speaking to reporters, acknowledged that the legislation he’s advancing 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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