GOP Sen. 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 (Texas) said he and key Democratic senators are trying to negotiate a deal to vote on a slate of gun control measures this week.

"We're talking to Sen. Murphy, and others, Sen. Blumenthal, there's some other things they said they would like to have a vote on so we're trying to figure out what those are and if it's possible to come up with a reasonable list of four, you know two on each side, things we could vote on to get this done ... this week," he said.

Cornyn added that both sides have agreed to check back in with each other after closed-door caucus lunches on Tuesday, when the parties will likely discuss their strategy.

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In addition to needing a deal as a group, any push to quickly schedule and speed up gun control votes in the Senate would require the consent of every senator — potentially a herculean lift on such a thorny issue

Talks of trying to have votes in the Senate this week comes as lawmakers are under growing pressure to pass new legislation amid a renewed national debate after this month's Parkland, Fla., shooting, where 17 people were killed.

One of the votes, if senators can get a deal, would be on the Fix NICS (National Instant Background Check System) Act that was spearheaded by Cornyn and Murphy.

The legislation would reinforce existing laws by ensuring that authorities report criminal records to the NICS and penalizing agencies that don’t provide the information to the FBI.

He added that Democrats will push for universal background checks, including closing the "the gun show and internet sales loopholes that allow guns to fall into the wrong hands."

A wide-ranging gun control debate isn't without risks for Senate Democrats, who are defending 10 seats in red and purple states carried by Trump in 2016.