Senators trying to lock down deal on gun votes this week
© Greg Nash

GOP Sen. John CornynJohn CornynTrump adds to legal team after attacks on Mueller Senate tees up Yemen vote for Tuesday Senate GOP: Legislation to protect Mueller not needed 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. 
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.

A spokesman for Murphy didn't immediately respond to a request for comment about what other measures the Democratic senator, who has been a leading advocate on gun control, would want a vote on. 
"If all Congress does in response to the Parkland shooting is to pass the Fix NICS bill, it would be an abject failure and a dereliction of our duty," Schumer said. 

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.  
Meanwhile, GOP leaders in the House are preparing a legislative package aimed at bolstering school safety in reaction to the Florida high school shooting.