Washington (CNN) In a week the Senate is supposed to debate and vote on major immigration legislation for the first time in years -- and only one person might know what it will look like: Senate Majority Leader Mitch McConnell.

"That, you'd have to check with the leader on," said Sen. Cory Gardner, R-Colorado, Monday about the process as he left a GOP Senate leadership meeting.

"You'll have to ask him," echoed fellow leadership member Sen. John Thune of South Dakota. "He'll have to decide what he wants to do."

"Sen. McConnell hasn't announced his intention," Majority Whip John Cornyn told reporters.

Lawmakers of both parties told reporters Monday repeatedly they had no idea what the legislation or the process they'd be voting on likely next week would look like.

McConnell promised to turn to immigration on the Senate floor after February 8, the next date that government funding runs out, if broad agreement couldn't be reached in that time. The promise, which he made on the Senate floor, was instrumental in ending a brief government shutdown last month , with senators of both parties pointing to the pledge for a "fair" floor debate as a major breakthrough.

The reality is, though, that McConnell has a lot of discretion as to how such a vote could go -- and as of now, he has not given many clues.

Even in a meeting with White House chief of staff John Kelly, Homeland Security Secretary Kirstjen Nielsen and White House legislative director Marc Short, a source said McConnell "wouldn't indicate what he's going to do."

"Total poker face," the administration official said. "He's not going to tip his hand."

But the group came for the meeting, the official said, to "make sure he hears from the administration."

On Monday, lawmakers expressed hope that such a deal could come together before the Thursday funding deadline, but wouldn't call it likely. That tees up a vote next week with an uncertain end.

"Probably if nothing is agreed on this week, which I would not be optimistic will happen, then Mitch'll call up some bill next week and let everyone get their votes on their amendments and see where it goes," Thune said. "My assumption is that in the end, something will pass. But I guess we'll see."

McConnell's choices will be instrumental in deciding how the debate goes, lawmakers and experts say, and he has a number of options on how to proceed, from the base bill, to the amendment process.

"There's a lot of different conversations that continue, I don't think anyone has narrowed it down to one, two or even three paths at this point," Gardner said.

Utah Sen. Orrin Hatch, the longest-serving GOP senator, was the only lawmaker who seemed to know how the debate would look.

"I have a pretty good sense. I've been through it a hundred times," he said, laughing. Asked if that meant a mess, he added, still chuckling: "It's always a mess."

What is the base bill?

Lawmakers have been lobbying McConnell in a number of different directions on what to actually call to the floor, which will form the starting point for a bill that will ultimately need 60 votes and bipartisan support to advance.

Some, like Oklahoma Republican Sen. James Lankford, argue the bill should reflect what President Donald Trump has proposed -- including elements Democrats say are virtually unworkable for them like cutting family-based migration and ending the diversity visa lottery.

After the meeting both Short and Nielsen also indicated to reporters the administration prefers McConnell to call the President's proposal.

Others, like Thune and West Virginia Democrat Sen. Joe Manchin, have suggested the framework should be far narrower, perhaps only border security funding and a permanent future for young undocumented immigrants who came to the US as children.

Manchin, who has been co-leading a group of roughly 20 bipartisan senators meeting virtually daily to try to find common ground on the issue, said the base bill will dictate the direction of the debate and how amendments should be handled.

"I think the base bill will tell you everything," Manchin said. "If it's going to be a neutral bill -- a neutral bill says it has a little bit, but not enough, to do everything everybody's wanting to do."

The No. 2 Senate Democrat Dick Durbin said McConnell had yet to start negotiating with Democratic leadership about what a bill will look like -- a step that may be necessary to get Democrats' agreement to even open debate on whatever the underlying bill is.

"Not at all," Durbin told CNN of any outreach. "He hasn't told us the base yet. All he's told us is 'level playing field.'"

Amendment process unclear

Beyond the base bill, McConnell will also set the tone for the amendment process.

Lawmakers have said the majority leader pledged an "open" amendment process -- which would suggest very few limitations on number of amendments and subject them to a 50-vote threshold. But McConnell never actually said the word "open" in his pledge, only committing to "an amendment process that is fair to all sides."

The indication is that amendments will likely require 60 votes, similar to advancing legislation, which would force any change to the underlying bill to reach a high, bipartisan bar.

"We'll see who can get to 60 votes," McConnell said at a Republican retreat last week. When asked if McConnell was referring to amendments as well, McConnell spokeswoman Antonia Ferrier told CNN the floor process was still not decided but "60 votes is a reality in the Senate" and "is a reality on amendments as well."

"I think you have to be at 60 on everything, if that's the way you're going to do it. If that's the end (goal)" Thune said, adding there would likely be a "reasonable number of amendments" for both sides. "Hopefully Democrats will be willing to enter into some sort of time agreement on amendments so it doesn't become an exercise with no end."

Durbin said 60 votes on amendments is "probably inevitable" but said he's not sure Democrats will go for substantially limiting the number of amendments that can be offered.

"He didn't say that," Durbin said of McConnell limiting what's called the tree of amendments. "Open amendment process so both sides can work their will. I would be for a more open process with 60-vote margins -- I think that reflects reality."

But leaders would likely face some pushback on that proposal. A source familiar with the thinking of the Senate Steering Committee, a conservative caucus led by Utah Sen. Mike Lee that advocates for open Senate processes, rejected the notion that being too open with amendments would be a free-for-all.

"I think free-for-all is supposed to be what we're about," the source said. "It's what Senate rules are supposed to allow. I'd much rather see that than the opposite of that."

James Wallner, a senior fellow of the R Street Institute and former executive director of the Senate Steering Committee under Republican senators, said both parties tend to structure process for bills like this to leave room to point fingers when it all falls apart.

"I can't even remember the last time the Senate had an open process," Wallner said. "McConnell, temperamentally, he's not one to let go and let it play out however it plays out. He likes to control things, and my guess is he will structure the floor process in some way that allows him to exert some level of control."