Sen. Chris Murphy, D-Conn., also part of the bipartisan talks, tweeted Wednesday there is \"growing momentum\" on a bill that can get \"broad bipartisan support in the Senate\" and that the group will keep working, but the group has remained tight-lipped on what the final proposal will include. Murphy has acknowledged it wouldn't include an assault weapons ban, as it wouldn't garner enough Republican support, despite similar legislation passing in 1994.

\"PHOTO:
J. Scott Applewhite/AP
PHOTO: Sen. Chris Murphy, the Senate's staunchest gun control advocate, speaks to activists demanding action on gun control legislation, the wake of the shooting at a Texas elementary school this week, at the Capitol in Washington, D.C., May 26, 2022.
>

The House Judiciary Committee convened Thursday for a markup on a package of gun control measures, called \"Protecting Our Kids Act,\" paving the way for the full House to approve the proposals as early as next week -- and then to send them to the Senate. It would raise the age limit for purchasing semiautomatic rifles to 21 years old, make it a crime to import or manufacture high-capacity magazines and further tighten the regulation of unregistered \"ghost guns\" and \"bump stocks,\" while also incentivizing the safe storage of firearms.

Democrats and Republicans argued bitterly over the proposed gun control measures, with both sides accusing the other of politicizing the latest slew of mass shootings across the country.

Democrats angrily denounced Republicans for repeatedly opposing gun control laws that could deter and prevent future mass shootings.

\"The public is demanding that we take action,\" Rep. Steve Cohen, D-Tenn., said.

Rep. Lucy McBath, D-Ga., whose 17-year-old son was shot and killed in 2012 over playing loud music, urged lawmakers to \"imagine the phone calls parents in Uvalde received last week\" from authorities when their children were killed.

\"I know that phone call, parents across the country know that phone call. It's a sucker punch to my stomach every time I learn there's another phone call. a phone call that brings you to your knees when the desperation will not let you stand, that leaves you gasping for air,\" she said.

Rep. Jim Jordan, R-Ohio, seized on Biden's recent comments criticizing the use of 9mm handguns and ammunition -- an aside that has spread across conservative media -- suggesting it was evidence that Democrats want to repeal the Second Amendment.

\"It's likely just the start,\" he said of the package, which also includes a measure to incentivize the safe storage of firearms.

\"The bill that Democrats are putting forward today does not really help us understand what drives some young men to commit these heinous acts,\" he said.\" It's a one-size-fits all approach that punishes law abiding citizens while doing nothing to make our committees safer,\" Jordan said.

Rep. Greg Steube, R-Fla., appearing virtually from his home near Tampa, pulled out several pistols that he claimed would be outlawed under Democrats' proposal.

\"I hope that gun is not loaded,\" Rep. Sheila Jackson Lee, D-Texas said.

\"I'm in my house I can do whatever I want,\" he replied.

Notably, the House has already passed gun control measures in this session of Congress, but that legislation has stalled in the Senate given GOP opposition and the 60-vote threshold -- so the real fate of gun control reform in the U.S. now mostly lies with the Senate group talks.

Still, amping up the pressure, House Speaker Nancy Pelosi at an event in California on Wednesday suggested Democrats would also consider holding a full House vote on an assault weapons ban following the recent mass shootings -- a non-starter for Senate Republicans.

\"As we get through those we will have hearings and marking up the assault weapons ban,\" Pelosi said. \"We are just trying to hit it in every possible way.\"

While the House measures wouldn't overcome a GOP filibuster in the Senate, the action is meant as so-called \"political messaging\" to pressure Senate Republicans and may put further pressure on negotiators to reach an agreement on areas of potential compromise.

\"PHOTO:
J. Scott Applewhite/AP
PHOTO: Senate Minority Leader Mitch McConnell walks to the chamber on the morning after an 18-year-old gunman opened fire at a Texas elementary school, killing at least 19 children, at the Capitol in Washington, D.C., May 25, 2022.
>

Republican Leader Mitch McConnell tasked John Cornyn, R-Texas, to negotiate with the group of senators, but said at a press event in Kentucky Wednesday that \"hopefully\" senators will \"find a way to come together\" to make progress on gun violence.

\"It seems to me there are two broad categories that underscore the problem: mental illness and school safety,\" McConnell said. \"So hopefully we can find a way to come together and make some progress on this horrendous problem consistent with our Constitution and our values.\"

As lawmakers seek compromise, it appears the American public is widely supportive of universal background checks and red flag laws. An ABC News/Washington Post poll conducted in September 2019 found 89% support for universal background checks and 86% support for red flag laws. Mandatory background checks and red flag laws also won support from at least eight in 10 Republicans and conservatives, the poll found.

The full Senate and House are scheduled to return to Washington next week.

ABC News' Allison Pecorin and Benjamin Siegel contributed to this report.

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