Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellOcasio-Cortez to voters: Tell McConnell 'he is playing with fire' with Ginsburg's seat McConnell locks down key GOP votes in Supreme Court fight Video shows NYC subway station renamed after Ruth Bader Ginsburg MORE (R-Ky.) is moving to banking reform legislation — not gun control or other responses to the high school shooting in Florida — next week in the Senate.

McConnell has filed a motion to have a procedural vote Tuesday on legislation sponsored by Senate Banking Committee Chairman Mike Crapo Michael (Mike) Dean CrapoBottom line Davis: The Hall of Shame for GOP senators who remain silent on Donald Trump Top GOP senator urges agencies to protect renters, banks amid coronavirus aid negotiations MORE (R-Idaho). After that, McConnell hopes to move to legislation addressing sex trafficking, according to GOP sources.

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Legislation addressing the Florida high school shooting, the subject of contentious conversations between President Trump Donald John TrumpBiden leads Trump by 36 points nationally among Latinos: poll Trump dismisses climate change role in fires, says Newsom needs to manage forest better Jimmy Kimmel hits Trump for rallies while hosting Emmy Awards MORE and GOP lawmakers at a White House meeting televised live on cable news Wednesday, will wait.

A Senate GOP aide said a limited bill to strengthen background checks for firearms purchases could come to the floor at any moment as soon as Democrats agree to let it move forward.

McConnell on Tuesday blamed Democrats for preventing the proposal, sponsored by Senate Republican Whip John Cornyn John CornynCalls grow for Biden to expand election map in final sprint Bipartisan praise pours in after Ginsburg's death Chamber of Commerce endorses McSally for reelection MORE (R-Texas) and Democratic Sen. Chris Murphy Christopher (Chris) Scott MurphyDemocratic senator calls for 'more flexible' medical supply chain to counter pandemics The Hill's 12:30 Report - Presented by Facebook - Don't expect a government check anytime soon GOP chairman to release interim report on Biden probe 'in about a week' MORE (Conn.), from coming to the floor.

“We tried to get it cleared yesterday, but the Democratic leader objected,” McConnell said.

Democrats vigorously dispute that and say that conservatives led by Sen. Mike Lee Michael (Mike) Shumway LeeMcConnell shores up GOP support for coronavirus package McConnell tries to unify GOP Davis: The Hall of Shame for GOP senators who remain silent on Donald Trump MORE (R-Utah) objected to moving the Cornyn-Murphy bill.

Either way, that legislation is not seen as a huge step on gun violence by most lawmakers. It would give local and federal authorities more incentive to report relevant information to the National Instant Criminal Background Check System (NICS). Trump on Wednesday suggested that additions should be made to the bill as he spoke with Cornyn.

Yet even that legislation has its opponents in the Senate.

Conservative Sen. Rand Paul Randal (Rand) Howard PaulSecond GOP senator to quarantine after exposure to coronavirus GOP senator to quarantine after coronavirus exposure The Hill's Morning Report - Sponsored by National Industries for the Blind - Trump seeks to flip 'Rage' narrative; Dems block COVID-19 bill MORE (R-Ky.) on Thursday said he had serious due process concerns over the potential that veterans diagnosed with post-traumatic stress symptoms could be blocked as a group from buying firearms.

“You can’t take gun rights away in bulk. If you say everyone that has PTSD that’s a veteran, all their data will be dumped into a database and it will show up on a background check, that’s a problem,” he said.

“I’m for taking away gun rights from violent people but you have to do it one at a time, you can’t do it in bulk,” he said.

Senate Democratic Leader Charles Schumer Chuck SchumerRepublican senator says plans to confirm justice before election 'completely consistent with the precedent' Video of Lindsey Graham arguing against nominating a Supreme Court justice in an election year goes viral Graham signals support for confirming a Supreme Court nominee this year MORE (D-N.Y.) says he supports the Fix NICS legislation but warned earlier this week that it falls far short of what is needed to stop mass shootings such a the one that left 17 people dead at Marjory Stoneman Douglas High School in Florida.

“If that is all Congress does, we won’t have done our job to keep America’s families safe,” he said.

He called the bill “fine” but “certainly not enough.”

Republicans are divided over what to do on gun legislation.

Some Republicans such as Lee, Paul and Sen. John Kennedy John Neely KennedyMORE (La.) say the Fix NICS bill is flawed.

Other Republicans, such as Sen. Jeff Flake Jeffrey (Jeff) Lane FlakeJeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Republican former Michigan governor says he's voting for Biden Maybe they just don't like cowboys: The president is successful, some just don't like his style MORE (Ariz.), want to go further by also raising the age requirement for buying rifles from age 18 to 21.

Sen. Pat Toomey Patrick (Pat) Joseph ToomeyAppeals court rules NSA's bulk phone data collection illegal Dunford withdraws from consideration to chair coronavirus oversight panel GOP senators push for quick, partial reopening of economy MORE (R-Pa.) thinks his bill co-sponsored with Sen. Joe Manchin Joseph (Joe) ManchinNames to watch as Trump picks Ginsburg replacement on Supreme Court Momentum growing among Republicans for Supreme Court vote before Election Day Gardner on court vacancy: Country needs to mourn Ginsburg 'before the politics begin' MORE (D-W.Va.) to extend mandatory background checks to include sales at gun shows and over the internet, should get priority.

Many Republican senators, however, say they oppose rising the mandatory age for buying rifles or requiring background checks for gun shows and online sales.