A proposal for expanded background checks on firearms by Attorney General William P. Barr left senators with more questions than answers Wednesday as they wait for President Trump to propose legislation aimed at cutting gun violence.

“I’m just not sure where we’re at,” Sen. Chris Murphy, Connecticut Democrat, told reporters. “Are we supposed to negotiate over this? Are we supposed to wait until the president signs off on it?”

Mr. Barr visited Capitol Hill on Wednesday to talk with lawmakers about expanding background checks and floated a proposal to do so.

The document, first reported by The Daily Caller, said background checks would be extended to include all commercial sales, including gun shows.

The checks would be conducted through a Federal Firearm License or a newly created class of license transfer agents. Sellers could keep their own records of the sale, though they could allow the FFL to store them.

While it was the first step in testing a position for Mr. Trump and the Republicans, several senators said they needed more details before they could decide whether to support it.

Their questions included details on how the Federal Firearm Licensee system would be overhauled and operated, who would keep records of sales, and what enforcement penalties would be in place.

When asked about the new class of licensing agents, Sen. Josh Hawley said he couldn’t describe it because there was no text to work with.

“It’s hard to really nail it down,” the Missouri Republican said.

Despite the questions they had about the proposal, senators said it was a “constructive” starting point.

But the biggest question remained: Where does the president stand?

“That’s an important piece — if the president doesn’t support it, there’s no point. It’s not going to become law,” Mr. Hawley said.

It was not clear what Mr. Trump thought of the proposal, with lawmakers pointing out that he hasn’t signed off on it and Mr. Barr was taking a temperature check.

He did, however, fire a warning shot Wednesday, slamming Democratic presidential contender Beto O’Rourke for proposing a mandatory buyback plan for AK-style guns.

“Dummy Beto made it much harder to make a deal,” Mr. Trump tweeted. “Convinced many that Dems just want to take your guns away. Will continue forward!”

While Senate Republicans mulled background checks, their colleagues in the House pushed back against the notion.

At a press conference Wednesday, Minority Whip Steve Scalise and several other House Republicans said Democrats were rushing to push through ineffective solutions.

“Unfortunately, whenever we hear about a tragedy, before we even know the facts, there are people right here on Capitol Hill that are rushing to find a microphone not to pray for the victims but to promote their own gun control agenda,” Mr. Scalise said.

Rather than focus on expanding the national background check system, which they argued doesn’t work, the House Republicans argued the real solution would be in addressing the factors that are breaking down society and in restoring family and religious institutions.

Newly elected Freedom Caucus Chairman Andy Biggs shot down Mr. Barr’s proposal.

“I don’t know what the appetite is [in the GOP], but I personally wouldn’t support it,” he said.

It also didn’t receive any help from the National Rifle Association.

“This missive is a non-starter with the NRA and our 5 million members because it burdens law-abiding gun owners while ignoring what actually matters: fixing the broken mental health system and the prosecution of violent criminals,” Jason Ouimet, executive director of the NRA’s Institute for Legislative Action, told Politico.

Democrats have pressed Mr. Trump and Senate Republicans to pass the universal background checks bill that the House passed earlier this year.

Republicans have shot down that option, with several members saying they are concerned about how it would affect private sales between individuals.

House Speaker Nancy Pelosi and Senate Minority Leader Charles E. Schumer appealed directly to Mr. Trump on the phone Sunday.

“This morning, we made it clear to the president that any proposal that does not include the House-passed universal background checks legislation will not get the job done,” they wrote in a statement. “We will not stop until these bills are passed and our children’s lives are safe. We call upon Sen. [Mitch] McConnell to ‘Give Us a Vote!’”

On Wednesday, Democratic leadership didn’t say they would refuse to take up a weaker version of universal background checks, with both Mr. Hoyer and Democratic Caucus Chairman Hakeem Jeffries saying they would need to see what the Republicans offered before making a decision.

Rank-and-file Democrats agreed to wait and see rather than prepare to block a weaker version of their bill.

“I’m going to keep an open mind to see what they bring forward, but I agree with the speaker of the House in that if there are loopholes then people will find ways to get access to guns and we will not have delivered,” said Rep. Veronica Escobar, whose Texas community lost 20 people in the El Paso shooting.

However, they did say they expect the vast majority of their bill to be included.

Meanwhile, the momentum for any bill addressing gun violence remains with the president.

“Really I think the president needs to make a public declaration that this is what he is for or something else and I think that will clarify things and get us further down the road than we are now,” said Sen. John Cornyn, Texas Republican.

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