Originally Published: 09 JUN 22 09:19 ET

Updated: 09 JUN 22 10:09 ET

By Daniella Diaz, Sonnet Swire and Manu Raju, CNN

(CNN) -- Sen. Chris Murphy of Connecticut, the lead Democrat in the bipartisan negotiations on gun safety reform, told CNN he believes there will be more than 10 Republicans supporting gun safety reform in the Senate but a top Republican negotiator says no deal will come this week.

"I think we will put together a package that will get more than 10 Republican votes," he told CNN's John Berman on "New Day" Thursday.

Lawmakers have been facing intense pressure to act in the wake of recent mass shootings in the United States, and at least 10 Republicans need to vote with Democrats to clear the 60-vote threshold to break a filibuster and pass legislation. Murphy said he hopes the group can still reach agreement on a package by the end of the week.

"We're still driving towards that goal," he said. "This is complicated law, and we want to make sure that we get it right, not do it fast. So everybody is still at the table, nobody is walking away, and I'm still confident we have a path to get there."

He added: "Time is never our friend when we're talking about getting people to the table on something as difficult as gun laws and gun violence. I think that phenomenon likely still applies, except for the fact that something different is happening out there in the American public in both conservative states and more Democratic-leaning states you have parents that just are not going to accept nothing as an option this time."

Hours after Murphy said he hoped a deal would come this week, a top GOP negotiator poured cold water on those plans.

Sen. John Cornyn, chief GOP negotiator on gun legislation, told CNN "no," he doesn't expect a deal this week on guns.

"But having said that we're making good progress," he added.

The Texas Republican said he is optimistic a deal can be on the floor by the end of the month.

On what's being discussed, Murphy said federally mandated red flag laws are not on the table in discussions with Republicans. Those laws, which currently exist in 19 states, allow a judge to temporarily take away the guns of someone who is deemed a threat to themselves or others.

"I think there has been some lingering confusion," Murphy said. "Past proposals have suggested a federal red flag law. I've actually never thought that was a good idea. I don't think you want law enforcement to have to go into the federal courts to take, temporarily, firearms away from a dangerous individual. So we have to clean up some of the confusion around what we are proposing."

He added: "We are talking about is incentivizing state red flag laws and providing substantial funding to implement them."

He also said the group is "zeroing in" on the 18-21 age group on age limits or looking at juvenile criminal records for purchasing semiautomatic weapons.

"I think we continue to try to find a path to 60 votes that includes some provision that recognizes these 18- to 21-year-olds tend to be the mass shooters, and that many times they have juvenile criminal records or past histories of mental health that should prohibit them from buying a weapon," he said.

He added: "So we are zeroing in on this population, trying to find the path to bipartisan agreement. I will say I think there's Republican support for raising the age to 21. I don't know whether there are 60 votes for that proposal, so we're exploring some other important and impactful options."

He continued to be very optimistic about the process of passing gun safety reform in the Senate, after telling CNN Sunday that he was "more confident than ever" lawmakers will be able to get something done to address gun violence across the US. Murphy also met with President Joe Biden earlier this week as the negotiations intensified on Capitol Hill.

"The urgency from the American public doesn't seem to be going away," Murphy said Thursday.

The House voted Wednesday 223-204 to pass a wide-ranging package of gun control legislation called the "Protecting Our Kids Act." The measure is not expected to pass the Senate, however, amid widespread GOP opposition to stricter gun control.

Passage of the legislation in the House took place hours after an emotional hearing on gun violence in which families of victims pleaded for more action.