Tuesday's action comes as House-passed gun control measures to bolster background checks for gun purchases remain stalled in the Senate.
The Democratic-led House of Representatives Judiciary Committee voted along party lines to approve three measures: a so-called red-flag bill, a ban high-capacity ammunition magazines for consideration by the full House and legislation to prohibit people convicted of violent hate crime misdemeanors from possessing firearms.
"On the identical time, all of us wish to present protection to our huge Second Modification".
But McConnell insisted, as he has in the past, that he would not take up any legislation unless President Donald Trump agreed to sign it into law.
The bill has failed to make it to the president's desk even as Democrats ramped up called for gun control amid a slew of additional mass shootings.
But GOP arguments were countered by emotional appeals for gun control from Democrats, including Rep. Lucy McBath, D-Ga., whose son was shot to death in 2012.
"We'll see if we can come up with something that's acceptable to everybody", Trump said, adding that he plans to have another meeting on the subject tonight and speak with the senators again Thursday.
"We need to treat gun violence like the crisis it is", Newsom said in a statement.
The Senate's top Democrat - Sen. "Some things will never happen ... and some very meaningful things can happen", he said. "The idea of having background checks on commercial gun sales makes all the sense in the world and is very broadly supported".
While Graham did not explicitly say he'd back Manchin-Toomey now, he said it was a direction he was leaning.
Senate Majority Leader Mitch McConnell of Ky., arrives for a news conference following a Senate policy luncheon on Capitol Hill, Tuesday, Sept. 10, 2019, in Washington.
The White House had previously warned it would veto the Congress-passed background checks bill.
McConnell should quit stalling and "put the (background checks) bill on the floor" for a vote, Schumer said.
"I'm open to anything that we can get bipartisan consensus on that also makes sure that we're not overreaching and really beginning to threaten the rights of law abiding citizens of which the majority of people who own guns are", Tillis said.
It includes a so-called red-flag law aimed at making it easier for law enforcement to take away guns from those deemed risky by a judge; a measure barring people convicted of hate crimes from buying guns; and legislation barring, for civilian use, magazines that can hold more than 10 rounds.
"Over the August recess I had extensive conversations with my colleagues on both sides of the aisle, and with the White House, and I'm optimistic that we can reach agreement on a package would pass the Senate", Collins said.
McConnell reiterated on Tuesday that he would wait for the White House to propose gun legislation that Trump would sign.
However Toomey, Manchin and Murphy cautioned that they didn't win Trump's endorsement of their background examine invoice throughout their 40-minute phone dialog. He said he had discussed the matter with the President on Wednesday evening.
Manchin, referring to Trump, said, "We're going to know hopefully by tomorrow if there's something we can all agree on".