(CNN) Key bipartisan senators worked to secure an agreement Tuesday that could allow for several votes on gun-related amendments possibly next week, even as most of the proposals being discussed face long odds in Congress.

The emerging agreement came as students from a high school in Florida where 17 people were killed by a gunman lobbied lawmakers to find a legislative response to the attack.

"I think we can work this," said Sen. Chris Murphy, a Democrat from Connecticut who was seen in a lengthy discussion on the Senate floor Tuesday with Majority Whip John Cornyn of Texas. "I hope Sen. McConnell is willing to have a structure debate on the floor that will allow us to see where the votes fall on a number of gun violence measures ... ultimately McConnell and Schumer have to come to that agreement."

Cornyn also said that they were trying to find a way to bring legislation that would incentivize state and federal agencies to enter more information into the country's background check database -- known as "Fix NICs" -- to the floor.

"It requires consent," Cornyn said, noting that the debate likely wouldn't begin until after this week.

Gun control efforts hit headwinds

Earlier in the day, it appeared Republicans on Capitol Hill were steering the debate away from proposals to restrict gun purchases toward plans to beef up school security after the latest mass shooting at a Florida high school.

While a group of students from Marjory Stoneman Douglas high school made the rounds pressing top leaders from both parties to move legislation to ban assault-style weapons and high-capacity ammunition clips, expand background checks and raise the age for purchases of rifles, there is no appetite in either the GOP-controlled House or Senate to move any significant gun control bills.

"We had a considerable discussion about the school safety issue at lunch ... the view of the vast majority of my conference is that we should make progress on bills that we agree on," Senate Majority Leader Mitch McConnell told reporters at the Capitol

House Speaker Paul Ryan stressed his long held position on Tuesday morning that Congress shouldn't be "banning guns for law abiding citizens." He said the appropriate response in the wake of the tragedy was to look at why the system failed to pick up warnings on the shooter and to make sure those with mental health or other issues don't get access to weapons.

The number three House GOP leader echoed the need to drill down on what went wrong in the response from police on the ground.

"As people are contemplating new laws, I think the most important thing we can look at is what about all the laws that are already on the books that were not enforced, that were not properly implemented? You know, I think what angered me the most is when I see breakdowns with law enforcement," said House GOP Whip Steve Scalise of Louisiana, himself a survivor of gun violence

Florida GOP Rep. Carlos Curbelo, who supports a measure to ban fire bump stocks and expand background checks, argued there were a lot of Republicans on Capitol Hill who would not be satisfied with narrow actions.

"It's not enough and there's a growing list of Republicans who agree it's not enough," he said. "This time has been different already, I can tell you just from many meetings and conversations."

Ryan and other leaders pointed out that the House already approved a narrowly created measure to patch up holes in the background check database, and urged the Senate to take action on that bill.

But one leading House conservative, Rep. Jim Jordan of Ohio, raised some concerns with that legislation, saying it lets bureaucrats take away Americans' Second Amendment liberties without a day in court.

"Here is what makes me mad is all these proposals don't address the problem," he said. "I mean there's was like 30 (times) this kid interacted with government. The day of the terrible attack, it looks like the sheriff and law enforcement didn't do their job so now the answer is more government?"

House Freedom Caucus Chairman Mark Meadows, a North Carolina Republican, said he and other conservatives are preparing to unveil a package of proposals to beef up security in schools that could cost billions.

He said the price tag for the proposal would be "substantial," noting that "it would start with a 'B,' not an 'M,'" and that "making our schools safe is something that I think all members of the Congress and the Senate are united in. How do you get there is a key question."

One idea conservatives are discussing is providing a tax credit for retired veterans or teachers to be trained to carry guns at schools to help respond to attacks. Another is working to provide greater support to law enforcement agencies.

"What we don't want it to do is to be a message," Meadows said. "We want it to actually be a results-oriented bill so normally when you do that it's significant dollars."

He added that there was close coordination with President Donald Trump's top advisers.

"I can tell you that the administration is engaged on a daily if not hourly basis, trying to get something done," he said.

More concerns from the Senate

Across the Capitol, Republican senators huddled for their weekly policy luncheon, but while members said they had a good discussion on guns, few seemed ready to act on any major proposals.

"I'm not sure there's anything we can do here that's going to prevent any of these things in the future, OK?" said Sen. Ron Johnson, a Wisconsin Republican.

Sen. Patrick Toomey, a Republican from Pennsylvania, who pushed for a plan to expand background checks back in 2013, said he'd had promising conversations with some members about garnering support for his bill again, but he wasn't definitive he'd be able to get the votes.

"I'm working on that," he said. "There are several senators who voted 'no' in the past who are open to reconsidering. Where they will end up, I do not know."

Republicans also seemed lukewarm on a plan floated by Trump last week to raise the age individuals have to be to buy a rifle. There appeared to be some openness to do that among a few Republicans like Sens. Marco Rubio of Florida and Pat Roberts of Kansas, but on Tuesday, Roberts didn't sound too optimistic.

"They backed off of that," Roberts said of the White House's position . "There's other things we should consider that would have a more dramatic effect."

It appeared the only thing that had some support was Cornyn's Fix NICs bill, but even that has hit snags on both sides of the aisle.

Democrats demand more

While many Democrats have supported the Fix NICs bill, which would incentivize federal and state authorities to enter more data into the country's background check system, Democrats argue it's not enough on its own.

"We want full debate. Not just on Fix NICs but on legislation that would really do the job," Senate Minority Leader Chuck Schumer said Tuesday. "The students say Fix NICs is not enough. The American people say Fix NICs is not enough ... we need full debate ... I was gratified to hear Sen. Cornyn say let's debate these issues. I'd like to hear Sen. McConnell say that as well. Because we have a moral imperative to act and act in a real way. Not just a little wink at the NRA."

If Democrats wouldn't support Fix NICs without universal background checks it could paralyze Congress from even passing the most narrow of the proposals they agree on.

"If we're going to fix NICs, what about really doing meaningful background checks?" said Virginia Democratic Sen. Tim Kaine.

Rep. Ryan Costello, a Republican from Pennsylvania, argued Congress needed to act even if it was just a small change.

"Passing a bill that is common sense even if it wouldn't have been the bill that would have prevented previous tragedy, isn't a reason not to pass the bill," Costello said.