WASHINGTON — House Democrats, seeking to seize the momentum after recent mass shootings and to pressure Republicans to embrace gun safety measures, will push forward on Tuesday with a new package of restrictions, including a bill that would ban the manufacture and sale of large-capacity magazines.

The House Judiciary Committee is expected to approve the package, the first step toward bringing it to the floor for a vote. It includes a so-called red flag law aimed at making it easier for law enforcement to take away guns from those deemed dangerous by a judge; a measure barring people convicted of hate crimes from buying guns; and legislation barring, for civilian use, magazines that can accept more than 10 rounds.

But the fate of the legislation — and of the broader effort to enact gun safety measures as massacres continue unabated — remains murky, with President Trump sending mixed signals about where he stands. Leading Republicans, who are reluctant to endorse any such measures without assurances that Mr. Trump is fully on board, are to meet with the president at the White House on Tuesday afternoon.

In the meantime, Democrats are making an intense push to pass their full wish list of gun safety measures, while also laying the foundation for making gun violence a central issue in their 2020 campaigns should a compromise falter.

Speaker Nancy Pelosi and Senator Chuck Schumer, the Democratic leaders, are demanding that Senator Mitch McConnell, the Republican leader, take up a House-passed bill extending background checks to all gun purchases, including those at gun shows and over the internet. Ms. Pelosi is also expected to appear Tuesday at a forum on gun violence legislation, convened by Representative Mike Thompson, Democrat of California, who heads a task force on gun safety.

“We will make this issue too hot for him to handle,” Ms. Pelosi said, referring to Mr. McConnell, at a news conference Monday afternoon, adding that gun safety groups “have upped the ante even further, to say that if this bill is not passed, Mitch McConnell and the Republicans in the Senate and the president will have hell to pay.”

The central figure in the debate, though, remains Mr. Trump. The president’s statements on gun safety have been “all over the lot,” Mr. Schumer said on Monday, and Mr. McConnell has said he will not bring any measure to the floor for a vote unless the president says he will sign it.

The president initially expressed support in early August for “very meaningful background checks” after a pair of deadly mass shootings in El Paso and Dayton, Ohio but stopped short of endorsing the bill passed by the House. Then, as has been his pattern in the past, Mr. Trump’s stance appeared to soften after talks with gun rights advocates.

The White House has been in talks with a bipartisan group of senators on a series of gun-related measures, including a red flag law and expanding background checks, as well as others measures that might be more palatable to the National Rifle Association. Senate Democrats involved in the talks said they expect White House aides to present Mr. Trump with his options as early as Wednesday, and remained unclear on what he might accept.

“It’s encouraging — they’re still talking,” said one of those Democrats, Senator Joe Manchin III of West Virginia, who is a chief sponsor of a background checks bill that fell to a Senate filibuster in 2013. That measure, which is slightly less restrictive than the House bill, would include certain exemptions for guns sold to friends and relatives.

But Senator Christopher S. Murphy, Democrat of Connecticut, who is also involved in the talks, sounded a note of pessimism, saying he feared that if Republicans or the White House demanded a package of measures — as opposed to a stand-alone vote on a background checks bill — the entire effort would collapse.

“I’ve expressed my worry to the White House the package could get so big that it could all fall apart,” he said.

Gun safety has burst onto the Washington agenda in the wake of a string of mass shootings in August — three in Texas alone. Democrats returning from their August recess say that it has been a huge issue in their districts, especially for young people and parents sending their children back to school.

“The number one item to purchase for your back-to-school kids should not be a bulletproof backpack,” Mr. Thompson said. Referring to the background checks bill that has not been brought up for a vote in the Senate, he added, “This is absolutely insane, and Senator McConnell can do something about it. “

Yet even as they push for measures beyond the background checks bill, Democrats are stopping short of their most ambitious goal: an assault weapons ban. A proposed ban is circulating in the House, but its chief sponsor, Representative David Cicilline of Rhode Island, estimated he is six or seven votes short of the 218 required for passage.

The Judiciary Committee will have a hearing on that bill later this month. Mr. Cicilline conceded that passage may be a heavy lift.

“Every other bill that we’ve done tries to keep guns out of the hands of people who shouldn’t have them,” Mr. Cicilline said. “This is the one piece of legislation that keeps a particular weapon out of the hands of law-abiding citizens. A lot of people have enormous objections to that.”

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