Washington (CNN) Discussions around legislative fixes to mass shootings heated up Wednesday as the attorney general returned to Capitol Hill to meet with lawmakers about proposed gun bills, including a measure that would expand background checks for buyers.

The pitch -- outlined in a document that was obtained by CNN -- would expand background checks to all advertised commercial sales, though it's not clear whether it would pass muster with lawmakers. The White House on Wednesday made clear that President Donald Trump had not signed off on any plan, and GOP leaders have indicated they are awaiting word from Trump before taking action.

The outline, which was first reported by the Daily Caller , is a Justice Department document.

In the wake of a series of mass shootings over the August congressional recess, there has been talk on Capitol Hill about what, if any, gun legislation could pass. But on the divisive topic of firearm regulation, Congress has repeatedly failed to enact stricter gun control in the aftermath of shootings, despite calls for action.

Senate Majority Leader Mitch McConnell has indicated that he doesn't plan to bring up any legislation in the Senate unless it is clear that the President would sign off on it -- and it's not yet clear what exactly the President would support.

White House deputy press secretary Hogan Gidley said in a statement on Wednesday, "The President has not signed off on anything yet but has been clear he wants meaningful solutions that actually protect the American people and could potentially prevent these tragedies from ever happening again."

Attorney General William Barr was on the Hill for the second day in a row, meeting this afternoon with lawmakers alongside the top legislative affairs aides at the Justice Department and the White House, a Justice Department official told CNN.

According to a copy obtained by CNN, the document circulating on the Hill describing the proposal states that "a background check requirement would be extended to all advertised commercial sales, including sales at gun shows," and that it would follow in the mold of background check legislation introduced by Republican Sen. Pat Toomey of Pennsylvania and Democratic Sen. Joe Manchin of West Virginia in 2013.

White House aides have spent the last month meeting with congressional staffers and devising a legislative measures. The Department of Justice also prepared a package of options that was delivered to the White House more than two weeks ago.

But the White House has yet to publicly release any gun measures that it would support.

Some of the options being proposed in discussions, in addition to measures Trump and his aides have floated publicly, include allowing minors' records to be included in background check databases, alerting local authorities when someone fails a background check, applying bigger penalties for straw purchases (when someone buys a gun for someone else) and helping states implement "red flag" laws, which would remove weapons from people deemed at risk.

Barr has been supportive of expanded background checks in conversations with the White House, CNN has reported. The gun package has become a top issue for the attorney general, who has explored the policy questions in a number of meetings at the Justice Department with senior law enforcement officials, including from the FBI and the Bureau of Alcohol, Tobacco, Firearms and Explosives, the Justice Department official said.

Republican lawmakers react to background check proposal

When asked if he thinks the Barr proposal has a future in the Senate, McConnell told CNN, "As I've said repeatedly, we are still waiting a proposal from the White House and I know they are considering everything. And since he's part of the administration, I'm sure what he's talking about is part of their consideration."

In one indication that it may be difficult to win a critical mass of Republican support, the National Rifle Association said in a statement that it opposes the document being circulated on the Hill.

Jason Ouimet, executive director of the NRA's Institute for Legislative Action, said in a statement, "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."

The proposal met with praise and skepticism in a Senate Republican closed-door policy lunch Wednesday, according to participants who said they held a lengthy discussion about all the latest gun proposals to see if they could unify around any of them.

"It was a family discussion," said Republican Sen. Kevin Cramer of North Dakota, who insisted it wasn't heated but "very respectful."

Leaving the lunch, most senators avoided reporters' questions about the proposal by saying that they had not had time to study it closely.

But Cramer, a conservative, said the proposal is "most closely aligned" with the plan from Manchin and Toomey that would expand background checks to all commercial sales.

Cramer described Manchin-Toomey as "probably the furthest out from the Republican dogma" and something his constituents consider to be a "slippery slope and probably something that is not helpful."

The comparison to Manchin-Toomey could spell trouble for the latest proposal because that bill wasn't able to pass the Democratic-controlled chamber in 2013 and is opposed by most Republicans.

As he headed into the lunch, Toomey said he had spoken to Barr about the proposal and that the attorney general has "some very constructive ideas. Creative and thoughtful. In all likelihood, I would be very supportive. There are some details we need to flush out."

Republican Sen. Ted Cruz of Texas, refused to embrace the proposal, saying the Senate should instead pass a proposal from him and GOP Sen. Chuck Grassley of Iowa that aims to tackle the mental health component of the mass shootings crisis and beef up prosecutions of gun crimes.

"I believe that the proper path for the Senate to take is to vote on Grassley-Cruz and pass it," he said.

Cruz said the entire debate had been poisoned by Democratic presidential candidate Beto O'Rourke's pronouncement at the last debate that if he were president he would want to confiscate weapons.

Sen. John Kennedy, a Republican of Louisiana, who said he has a meeting later in the day with Barr, insisted the proposal "hasn't gelled yet."

"I think this is still a very fluid process. I could be wrong but I doubt it," he said. "I don't think the White House has definitely decided on a way forward and we're talking about it."

Sen. Ron Johnson, a Republican from Wisconsin, said the White House and Senate Republicans are giving "a great deal of thought" to the various "proposals that are meaningful and would make a difference."

"That's why it's taking some time," he said.

This story has been updated with additional developments Wednesday.