Senate Democrats offered their ante Thursday in the renewed gun control debate, saying any bill must expand background checks and allow court-ordered gun confiscations while demanding a vote on a semi-automatic rifle ban — though they said that is not a deal-breaker if it doesn’t become law.

Senate Minority Leader Charles E. Schumer of New York said his Democratic caucus is prepared to provide a “very large” number of votes for those priorities. He called for a free-ranging debate on the chamber floor, where he said he would accept the outcome.

“This is what we think should be done. This is the right thing to do … but if we can get a good portion of it and not all of it, fine,” Mr. Schumer said. “To get these things done, I’m not drawing any lines in the sand.”

Democrats sense they may have gained the upper hand in the debate after a stunning meeting Wednesday between members of Congress and President Trump at the White House, where Mr. Trump shot down several Republican priorities and accused lawmakers of being “petrified” of the National Rifle Association.

Mr. Schumer said he is counting on Mr. Trump and that the extent of a final bill will depend on how far the president is willing to push members of his own party.

“The only hope of passing this, given that Democrats are so strongly for these proposals, is the president persuading Republicans and frankly giving them cover from the NRA,” he said.

After years of stalemate, the debate in the wake of the Valentine’s Day massacre at Marjory Stoneman Douglas High School in Parkland, Florida, feels different.

Police say Nikolas Cruz, the 19-year-old accused of slaying 17 people, showed warning signs that he wanted to hurt people with guns before he carried an AR-15 rifle into his former high school on Feb. 14.

Mr. Schumer said Democrats will push for protective court orders that would allow law enforcement or family members to petition judges to block people from getting guns if they see danger signs.

Mr. Schumer said Democrats’ big goal is expanding gun purchase background checks to cover private sales at gun shows or online. These sales currently are exempt from federal requirements. A more modest background check measure that encourages record-sharing, which has wide bipartisan support, isn’t enough, he said.

“Rather than just plugging one leak, we need a holistic approach that improves our gun safety laws from top to bottom,” he said.

Mr. Schumer also said Democrats want a debate on the Senate floor on a so-called assault weapons ban, though he acknowledged it would be unlikely to pass, with opposition from members of his own party as well as Republicans.

House Minority Leader Nancy Pelosi, California Democrat, said her troops also will embrace gun violence restraining orders and comprehensive background checks. She said an assault weapons ban could take longer.

“We need to have the best package we can get done now,” Mrs. Pelosi said.

Some senators pushed for a debate this week, but the Senate spent its time jumping through Democrat-imposed procedural hoops to confirm some of Mr. Trump’s nominees. Senate Majority Leader Mitch McConnell, Kentucky Republican, said a banking bill will be on the floor next week.

Lawmakers also are staring at the next government funding deadline on March 23, which could further delay a potentially divisive gun debate.

While Democrats show significant unity as they approach the debate, Republicans are fractured.

Sen. Marco Rubio, Florida Republican, said Thursday that he will introduce a version of the gun violence restraining order and that he wants to increase federal funding for school safety initiatives and crack down on people who purchase weapons illegally.

He said he would continue to look into new age restrictions for buying certain guns and limits on ammunition magazine capacity, but he added that those ideas aren’t as likely to attract widespread support in Congress.

“Do not hold hostage a piece of legislation that would work and that we all support because it doesn’t have everything you want,” Mr. Rubio urged his colleagues in a floor speech.

Sen. Jeff Flake, Arizona Republican, along with Sen. Martin Heinrich, New Mexico Democrat, and Sen. Catherine Cortez Masto, Nevada Democrat, introduced a bill Thursday to prohibit the sale of “bump stocks” — devices that convert a semi-automatic shotgun into a fully automatic weapon that gained attention after the Las Vegas shooting last year.

Mr. Trump has ordered Attorney General Jeff Sessions to craft a ban, but lawmakers have pointed to multiple rulings during President Obama’s administration that the Bureau of Alcohol, Tobacco, Firearms and Explosives lacks the capacity to regulate many of them.

“A temporary regulatory fix is no substitute for permanent law,” Mr. Flake said.

Mr. Flake is part of bipartisan Senate groups pushing legislation to raise the minimum age to purchase certain rifles from 18 to 21 and to prevent people on “no fly” or terrorism watch lists from buying guns.

Sen. Patrick J. Toomey, Pennsylvania Republican, and Sen. Joe Manchin III, West Virginia Democrat, are also co-sponsors of the “no fly, no buy” legislation. Mr. Toomey said they are eyeing a new version of the bipartisan legislation they introduced in 2013 that would expand background checks to gun sales at gun shows and online.

Mr. Trump said the Toomey-Manchin legislation could serve as the basis for a more comprehensive bill, though the White House seemed to walk that back Thursday.

“I can’t say where we fall in position on that piece of legislation at this point,” said White House press secretary Sarah Huckabee Sanders. “He’s continuing to have conversations.”

• Dave Boyer contributed to this article.

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