A group of key senators plans to meet Tuesday to discuss a possible bipartisan gun control compromise, at the beginning of what might be the most crucial week for gun control legislation in decades. 

"I've seen more Republican interest in coming to the table and talking this time than at any other moment since Sandy Hook," Sen. Chris Murphy, D-Conn., said on CBS' "Face the Nation" this week. "It is true, you know, Republicans are not willing to support everything that I support, like banning assault weapons. But I really think that we could pass something that saves lives and breaks this logjam that we've had for 30 years."

Murphy and Sen. John Cornyn, R-Texas, are leading preliminary talks, which come after 21 people, including 19 children, were gunned down last week at Robb Elementary School in Cornyn's home state. The discussions are likely to focus on possible modest legislation on background checks or encouraging state red flag laws. 

Sen. John Cornyn, R-Texas, questions Supreme Court nominee Ketanji Brown Jackson during a Senate Judiciary Committee confirmation hearing on Capitol Hill in Washington, Tuesday, March 22, 2022. (AP Photo/Andrew Harnik)

Sen. John Cornyn is leading a bipartisan Senate group considering a possible gun reform compromise. (AP Photo/Andrew Harnik)

Meanwhile, Democrats – deeply suspicious that there are enough GOP votes to get a deal through the Senate – are already preparing a slate of bills for votes in the House, and perhaps the Senate, to pressure Republicans to act. But Cornyn told reporters in Texas Monday he hopes there's room for some level of agreement, as long as both parties come to the table in good faith. 

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"I will do as I've always done, and that is try to lean forward and meet my colleagues across the aisle halfway, particularly on matters involving the seriousness and gravity that this does," Cornyn said at an appearance in San Antonio.

Sen. Chris Murphy reacts to the Texas school shooting

Sen. Chris Murphy speaks from the Senate floor after the school shooting in Uvalde, Texas. (Senate Pool Video)

"We're already having those discussions in person and on the phone and looking forward to meeting on Tuesday through a Zoom call to try to see if we can agree on a basic framework about how we go forward," Cornyn added. "One thing I hope does not happen is that the various parties sort of fall back into their talking points."

The meeting Tuesday is anticipated to take place at some point in the afternoon.

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Democrats view the next week, or perhaps two, as make or break for any bipartisan deal. They complain that despite several mass shootings in recent years, particularly since the Sandy Hook massacre in 2012, there's been a lot of talk about gun reform, but very little action. 

Murphy last week said he won't let Republicans drag out talks for an extended period of time if it's clear that a deal that can pass the Senate with 60 votes isn't likely. 

"I'm not gonna negotiate forever," Murphy told reporters last Thursday at a gun control rally outside the Capitol. "So if we can't get some progress by the end of next week, then I'll say to Sen. Schumer, it's time to take votes."

But any compromise is likely to face a difficult uphill battle. 

Uvalde Texas Robb Elementary school shooting memorial

Rene Lucero and his wife, Alva, visit a memorial at Robb Elementary School in Uvalde, Texas, on Tuesday, May 31. (AP/Jae C. Hong)

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Sen. Richard Blumenthal, D-Conn., said Thursday that there were six to 12 Senate Republicans seriously considering gun legislation, "beyond general expressions of interest." If that number does not increase significantly, the margin of error for Republicans who can back away from talks becomes extremely thin. 

A Senate Republican aide told Fox News Digital that the talks are a "good sign for bipartisan progress" and the fact that Senate Minority Leader Mitch McConnell, R-Ky., is supportive of them increases their chances for success. But, the aide said, "10 votes on the Republican side will be hard."

Another Senate GOP aide said leadership is giving room to the senators leading the talks but "it's all going to come down to the details… any deal that gets 10 Rs will require threading the needle." 

A third Senate GOP aide expressed optimism about the prospects for success of the negotiations, citing the fact that Cornyn and other Republicans with Second Amendment street cred are involved.

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"We are acutely aware that the overwhelming majority of gun owners are law-abiding," the aide said. "So we will be focusing on a few discreet areas where improvements can be made – not some comprehensive bill that will and up in a legislative trash heap."

Congressional building Capitol Hill

Democratic lawmakers say that in the wake of the Uvalde, Texas, school shooting, several Republicans appear open to supporting some form of gun legislation. (AP Photo/Patrick Semansky, File)

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But no matter what, Murphy told activists at the Capitol Hill gun control rally last week, the Senate will vote on gun bills. 

"We are going to force people to tell America which side they are on," Murphy said.

The House of Representatives, meanwhile, is not waiting on the Senate to act. The House Judiciary Committee is planning a meeting Thursday to prepare a slate of gun control bills for the House floor, Fox News confirmed. Punchbowl News first reported the news. 

Among the issues the bills will address will be an increase in the age for purchasing semi-automatic rifles from 18 to 21, banning new high-capacity magazines, tightening regulations on ghost guns and more. 

"Failure to take action to address gun violence after we witnessed a mass shooting that killed twenty-one people, including nineteen elementary school kids, is not an option," House Majority Leader Steny Hoyer told Fox News Tuesday. "I am encouraged by the bipartisan conversations in the Senate on legislation that could save lives by taking long-overdue steps to address gun violence. In addition, I have announced the House will take up legislation to create a national ‘Red Flag’ law to ensure those who pose an immediate threat to themselves or others are unable to possess a firearm." 

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Notably not included in the list of bills is a ban on assault rifles. Legislation on that may be tough to pass with Democrats holding a slim margin and many of them in close re-election races. A senior Democratic aide said conversations about future bills beyond what the Judiciary Committee is considering, including an assault weapons ban, are "ongoing" and they are "hopeful that every Member will see the urgency of this situation."

It's also highly unlikely that any of the legislation will get through the Senate. But quick action in the House – including a possible vote next week – may be viewed as a tool to pressure senators to come to some kind of agreement. 

Fox News' Lakita Atkins and Aishah Hasnie contributed to this report.