Congress appears to be settling on the outlines of consensus for action on stricter gun controls, with moves to strengthen the background check system and to restrict sales of bump stocks the most likely areas of bipartisan action.

Moves to raise the age limit on sales of rifles will test the limits of the deal, though, and even Democratic leaders sounded doubtful about their ability to ban sales of military-style semi-automatic rifles.

President Trump’s suggestion to train and arm school faculty volunteers willing and adept enough to carry firearms is also meeting with stiff resistance. The president now says he would leave the decision up to states.

Congress returns to Washington after a weeklong vacation Monday to find the debate has only intensified since they left, just a day after the Parkland, Florida, high school shooting that has ignited the latest national battle over gun control.

Led by Mr. Trump, Republican leaders are looking to speed through legislation that would require federal agencies to report more criminal and other disqualifying records to the National Instant Criminal Background Check System, which screens all gun purchases from licensed dealers. That could have prevented the gunman in the Nov. 5 Texas church shooting — who faced discipline in the Air Force for domestic violence — from obtaining weapons.

Key Republicans also say they are willing to consider legislation to outlaw bump stocks, which are aftermarket add-ons that make semi-automatic rifles simulate the rate of fire of automatic rifles. One was used in the Oct. 1 Las Vegas massacre.

Mr. Trump’s announcement that he wanted bump stock sales to be regulated may have cleared a path in Congress.

“With what the president proposed on bump stocks, it seems to me that is easier to deal with,” Sen. John Cornyn, Texas Republican, told The Washington Times. “It may be as simple as just providing explicit authority to the Bureau of Alcohol, Tobacco and Firearms to regulate bump stocks.”

Mr. Cornyn is the chief sponsor of the Fix NICS bill to add federal agency records and press states to ensure their records are part of the national background system.

That bipartisan bill is a likely starting point.

How much further the bill goes, though, depends on Mr. Trump and other Republicans.

Sen. Patrick J. Toomey, Pennsylvania Republican, said he intends to try again to expand background checks beyond licensed dealers to include sales between private parties online and at gun shows.

His proposal, co-sponsored with Sen. Joe Manchin III, West Virginia Democrat, failed to clear a Senate filibuster after the 2012 Sandy Hook Elementary School shooting, but there has been a new surge of support for it this month.

“Let’s at least require a background check for all commercial sales. … I intend to give this another shot,” Mr. Toomey told NBC’s “Meet the Press” on Sunday morning.

Mr. Trump and several other prominent Republicans have also backed raising the age for sales of rifles to 21, which would put them in line with the federal standard for sales of handguns. Some states already have a 21-year age limit for rifles.

Sen. Marco Rubio, Florida Republican, added his voice last week to those calling for a higher age.

But Mr. Cornyn said he opposes the idea.

“If you can be enlisted in the military at 18, I’m not sure I understand the 21 age. I think there are better ways to address it than just an arbitrary age increase,” he said.

Many of the students who survived the Marjory Stoneman Douglas High School shooting have called for a broad ban on sales of semi-automatic rifles such as the AR-15-style weapon that police say the killer used in that attack.

Sen. Dianne Feinstein, California Democrat, has said she would push for a renewed debate on such a ban, which was in place for a decade after the Clinton-era Federal Assault Weapons Ban. That law expired in 2004.

But Senate Minority Leader Charles E. Schumer, New York Democrat, seemed to downplay that proposal in a conference call with reporters last week. He said Democrats’ top priority going into the debate would be to impose universal background checks.

He said that was at “the nexus of the chance of actually becoming law … and at the same time doing a whole lot of good in preventing people — felons, those who are adjudicated mentally ill — from getting guns.”

Meanwhile, Mr. Trump’s call to have armed faculty available to deter or stop shootings has struggled to gain traction.

Florida Gov. Rick Scott, a Republican, became the latest to reject the idea on Sunday.

“I disagree with him. I believe you have to focus on the people who are well-trained, law enforcement, who are trained to do this,” he said on “Fox News Sunday.” “I want our teachers to teach, and I want our law enforcement to be able to protect the students. I want each group to focus on what they’re good at.”

But Rep. Brian J. Mast, a Florida Republican and former Army bomb disposal technician, said he could support armed teachers in some instances.

Mr. Mast said, though, that he has concluded there should be limits on the types of rifles available. He said Americans don’t need access to the kinds of weapons he carried during his time in the Army.

“I have fired tens of thousands of rounds through that rifle, many in combat. We used it because it was the most lethal — the best for killing our enemies,” he said in an op-ed Friday in The New York Times. “I cannot support the primary weapon I used to defend our people being used to kill children I swore to defend.”

He said Mr. Trump should use executive powers to declare a temporary ban on sales of AR-15-style weapons — similar to the president’s travel ban policy — giving Congress space to pursue a more permanent solution.

Rep. Thomas Massie, Kentucky Republican, said he would like to end gun-free zones because that’s where 98 percent of mass shootings happen.

“We need to take those labels off and put our kids in that 2 percent category of being safe instead of being in that 98 percent vulnerable category,” he said on NBC.

He also said he wasn’t a fan of raising the buying age.

“If I came home after proposing some of these things that are so unserious and disingenuous that some of my colleagues are proposing … I couldn’t face my wife or my children,” he said.

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