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A bipartisan group of senators Monday will meet in person on potential gun safety legislation, as the Senate inches toward a potential deal in the wake of several mass shootings

The meeting, confirmed by a source familiar to Fox News – with Chris Murphy, D-Conn., and John Cornyn, R-Texas, serving as the lead negotiators for their respective parties – follows a meeting between the group last Tuesday. That meeting was done via video conference as the Senate was out on recess.

A separate, larger group of senators also met last Wednesday to discuss areas of possible agreement between Republicans and Democrats. 

WASHINGTON, DC - AUGUST 04: Sen. John Cornyn (R-TX) speaks on a proposed Democratic tax plan, at the U.S. Capitol on August 04, 2021 in Washington, DC. The Senators spoke out tax proposal saying that it will hurt job growth and the middle class. (Photo by Kevin Dietsch/Getty Images)

Sen. John Cornyn speaks at the U.S. Capitol on Aug. 4, 2021, in Washington. (Kevin Dietsch/Getty Images)

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Sens. Kyrsten Sinema, D-Ariz., and Thom Tillis, R-N.C., will also join the meeting Monday, first reported by Politico. 

Murphy said Monday that while he is "prepared to fail," he is "more hopeful for success than ever before." 

"I'm trying to make clear that I'm not going to let the perfect be the enemy of the good," he added. 

Cornyn, meanwhile, said Monday that he won't sign onto any legislation 

"If we reach an agreement, law-abiding gun owners will not be impacted at all," he said on the Senate floor.

Talks on potential gun reform were spurred by multiple mass shootings last month, including one apparently racially motivated shooting at a Buffalo supermarket and another in an Uvalde, Texas, elementary school. 

Those shootings were followed by yet more major attacks, one at a Tulsa, Oklahoma, hospital and another over the weekend on South Street in Philadelphia, Pennsylvania. 

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The two most prominent potential areas for agreement in the Senate appear to be on legislation to encourage states to implement red flag laws and reforming background checks. 

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Democrats in the House of Representatives are also forcing votes this week on a number of bills, including provisions to ban high-capacity magazines and raise the age to own semi-automatic rifles. But that legislation will not pass in the 50/50 divided Senate, where 60 votes are needed to pass a bill. 

And even for a more modest Senate-negotiated package, a GOP aide told Fox News, any deal to get 10 Republicans and break a filibuster "will require threading the needle." 

Fox News' Kelly Phares contributed to this report.