The House has scheduled a vote next week on a bill to stop suspected terrorists from buying firearms from gun dealers as part of a new anti-terrorism package.

The gun ban is based on legislation written by Sen. John Cornyn, Texas Republican, that would grant the attorney general the power to halt gun sales for up to three days for anyone suspected of terrorist connections.

Authorities would then have to go to court and prove their case, or else the sale would go through.

Rallying his GOP troops before the vote, House Speaker Paul D. Ryan pitched the plan as a way to “step up our game” in the wake of a mass shooting at a nightclub in Orlando, Florida, that killed 49.

Gunman Omar Mateen pledged fealty to the Islamic State, and Republicans are trying to keep the spotlight on radical Islam, while Democrats point to lax gun laws as the main problem at home.

Mr. Cornyn’s proposal earned 53 votes in the Senate last month — shy of the 60 needed to overcome a filibuster, but more than any of the other alternatives, including a broader ban sponsored by Democrats and a narrow ban written by a bipartisan group led by Sen. Susan M. Collins, Maine Republican.

Republican leaders argue that the Democratic proposal and the Collins-led plan are both unconstitutional because they would strip people of their right to buy a firearm without due process.

The GOP leaders say both proposals put the burden on buyers, denying their rights based on secret FBI-maintained lists rather than on the government to prove terrorist connections.

House Majority Leader Kevin McCarthy announced the new bill Friday, saying it gives authorities new tools to combat radical Islam.

“It will also provide a process for individuals being investigated as known or suspected terrorists who attempt to buy a gun to be flagged, delayed and — if the burden of proof is satisfied — denied their purchase,” he said, referring to the Cornyn plan.

Democrats, however, have rallied to the Collins plan, with Senate Majority Leader Harry Reid declaring it a critical showdown that can break the power of the National Rifle Association.

House Minority Leader Nancy Pelosi called the Cornyn plan “a toothless NRA bill that will do nothing to keep our communities safe.”

“This bill is just the latest evidence that House Republicans have become a wholly owned subsidiary of the NRA,” Mrs. Pelosi said. “Democrats will continue to push House Republicans to give the American people a vote on meaningful gun violence prevention measures that will save lives and protect our communities from terrorism: with expanded, strengthened background checks and meaningful No Fly, No Buy legislation.”

The House was on vacation this last week, but in the prior week, Democrats shut down floor business for a day, conducting a 1960s-style sit-in to demand a vote on gun control legislation.

They carried on their protest in a series of rallies back home, linking arms with families of the victims of gun violence in sessions from California to Chicago to New Jersey.

In addition to the gun issue, the new House bill also would order the State Department to deny passports to anyone affiliated with groups the department has identified as terrorist organizations.

The bill appears designed to force a confrontation with President Obama, who has shied away from using the term “radical Islam,” saying he doesn’t want to conflate the terrorists with a religion. The House bill uses the term “radical Islamist” nearly 50 times.

“Let’s just let the facts speak for themselves. Al Qaeda and [Islamic State] are larger than they’ve ever been. We have attacks being conducted by people who are being radicalized over the internet in the United States of America and nearly every country around the globe. Europe’s being overrun with this problem,” Rep. Devin Nunes, California Republican and chairman of the House Permanent Select Committee on Intelligence, told “Fox News Sunday.” “So I just don’t think it’s helpful to not call it what it is.”

Incidents of Islamic radicalism mounted even as Republicans queued up next week’s votes. Machete-wielding attackers killed 20 hostages in Bangladesh on Friday, and a bomb blast killed more than 120 in Baghdad, Iraq, on Saturday, marking the third mass casualty attack in less than a week after a coordinated attack in Istanbul killed more than 40 at the Turkish city’s main airport.

The White House “strongly condemned” the Baghdad attack on Sunday and said it was committed to taking territory away from the Islamic State, though Republicans say the administration needs to beef up its strategy if it hopes to wipe out the group.

“We’re not responsible for the domestic security of every one of our allies,” Sen. Tom Cotton, Arkansas Republican, told NBC’s “Meet the Press.” “But ultimately, the way to stop attacks around the world is to deny terrorists the safe haven they need and to eliminate their leadership from the battle, just like we did with al Qaeda through much of the last decade.”

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