Washington (CNN) A bipartisan group of roughly 20 senators that's been huddling behind closed doors for weeks is furiously working to draft a bill that they can propose during an expected floor showdown on immigration next week.

If they are successful, it would mean at least one-fifth of the Senate would have established an influential voting block to shape the debate over immigration and the Deferred Action for Childhood Arrivals program

Emerging from one of their closed-door meetings Thursday, senators said multiple members are drafting language for compromise legislation, though they acknowledged they still don't have a consensus yet.

Sen. Claire McCaskill, D-Missouri, said she would be "shocked" if they didn't end up introducing their plan next week.

Republican Sen. Susan Collins of Maine, who hosts the meetings in her office on a near-daily basis, said there will "probably" be more than one proposal that emerges from their recent talks that could serve as amendments during next week's debate, though she added it's "too early to tell."

Senate Majority Leader Mitch McConnell is expected to bring up immigration next week in a rare neutral Senate floor debate. The Republican has pledged to allow for amendments from both sides, but it's still unclear how many amendments either side will be able to offer. And the expectation is any proposal would need 60 votes to succeed, a high bar that may make a major immigration compromise unlikely.

Other groups of senators are expected to introduce amendments as well. The White House also has its own framework, and Senate Majority Whip John Cornyn confirmed this week that some Republican lawmakers are working to draft a version based on those bullet points.

The bipartisan group of roughly 20 lawmakers, which calls itself the Common Sense Coalition, is aiming to operate as a voting bloc that can help steer the debate. Sen. Bob Menendez, D-New Jersey, who is working with members of the group said the goal would be to get 60 votes on the bipartisan amendment, and "then that would be it, we'd resist everything else, any other amendments."

It's unclear just how many members will make up the coalition in the end. The group could be influential if they vote as a unit, though it's not clear that everyone would get on board. Sen. Chris Coons, D-Delaware, said the number of supporters they have depends on the contours of the proposal. In their negotiations, sometimes a proposal will garner 30 members, while a different proposal will have 20 or 40.

"The challenge with immigration is it's a very broad range of concerns," he said.

Republican Sen. James Lankford of Oklahoma cautioned that a final deal hasn't been reached yet. "It's one thing to edit concepts, it's another thing to look at language and go, 'no this doesn't work,' and then try to make adjustments from there," he told reporters.

While the White House wants an immigration bill that focuses on four key pillars -- increasing border security, resolving DACA, ending the visa diversity lottery, and heavily curtailing family-based immigration, or chain migration, as they call it, multiple senators stressed that a bill has little chance of passing unless it narrows to just two of those pillars: DACA and border security.

"I think a lot of people are learning that immigration's complicated. The more we try to do, the more unanswered questions emerge," said Republican Sen. Marco Rubio, who was part of a different group of senators that pushed a much larger immigration bill in 2013. It was passed by the Senate but went nowhere in the House.

Like Rubio, Coons also endorses the concept of a narrower bill. "The challenge is, there's lot of other proposals that the White House and others want to address," he said.

The clock, however, is ticking, and the group is hoping to strike a final deal by Monday or Tuesday, roughly when McConnell is expected to kick off the amendment process.

"We don't have any choice, right?" said Sen. Jeanne Shaheen, D-New Hampshire. "Next week's coming."