Supporting the gun background check system, mental health resources, and security dollars for schools are just a few items that Republicans and Democrats are seeking common ground on for potential legislation, Murphy said Sunday.
Meanwhile, some Republican lawmakers said they won't back laws that limit gun rights.
"We're talking about red flag laws. We're talking about strengthening and expanding the background check system, if not universal background checks. We're talking about safe storage. And yes, we're also talking about mental health resources and more security dollars for schools. A package that really, in the end, could have a significant downward pressure on gun violence in this country," Murphy told ABC News's Jonathan Karl. "Maybe that's the most important thing we could do is just show that progress is possible and that the sky doesn't fall for Republicans if they support some of these common sense measures."
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Murphy added that negotiations have been conducted over the weekend with Sen. John Cornyn (R-TX), among other Democrats and Republicans, and are likely to resume through early next week.
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Senate Majority Leader Chuck Schumer turned to Murphy and Sen. Richard Blumenthal (D-CT), both of whom were involved in efforts to pass background check legislation after the shooting at Sandy Hook Elementary School in their home state of Connecticut in 2012, to represent Democrats in negotiations, giving them a 10-day deadline to reach a bipartisan agreement.
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Twenty-one people, including 19 children and two teachers, died in the shooting in Uvalde, Texas, on Tuesday. Law enforcement killed the gunman, identified as an 18-year-old who officials say legally purchased two AR-15-style rifles days after his birthday.
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Sen. Dick Durbin (D-IL) said he senses a "different feeling" among Senate colleagues that could prove promising for gun safety legislation.
"I can't say for certain, but I can tell you I sense a different feeling among my colleagues after Uvalde," Durbin, the Democratic whip, told CNN's Dana Bash on Sunday.
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Murphy sought to tamper with Democratic expectations, stressing that compromises would need to be made in order to pass any type of legislation.
"We've had a short timeframe, Jonathan. We've got to get this ready for Congress when Congress reconvenes in about a week, but I think we can do it," Murphy said.
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