Washington (CNN) Senate Republicans on Monday night declared they would only allow immigration to be on the Senate floor for one week -- pressuring Democrats to show their hand as the Senate inched closer to opening its debate.

Senators took a key step toward opening debate on immigration Monday evening, kicking off an exercise with little modern precedent that could affect millions of lawful and undocumented immigrants. The procedural vote, which passed 97-1, allows the Senate to next vote to officially open debate -- a vote expected on Tuesday.

But Senate Republican leadership said Democrats would have to act fast if they wanted to offer proposals.

"This is Sen. (Dick) Durbin and Democrats' opportunity and so far they kind of seem to be a little confused about what they're planning on doing -- but they better get it done quick because it's this week or not at all," said Senate Majority Whip John Cornyn.

"We need to get it wrapped up by Thursday," Cornyn said.

Just an hour previous, Cornyn had told reporters the opposite, saying: "We could do it this week if there is cooperation. If there is not, it might take longer."

A GOP leadership aide said Republicans want to light a fire under Democrats and get them to release their amendments. GOP leaders are worried Democrats want to drag out the debate for "weeks and weeks."

"That's the plan," said GOP leadership member Sen. John Barrasso when asked if this would be finished this week. He added that "Democrats have been waiting for this for a long time. They were promised we'd go to the floor with this and it's now on the floor."

Aides and lawmakers were unsure of any agreements on proceeding expeditiously. Without unanimous agreement from all senators, each amendment could take hours to consider. It was also unclear how far afield amendments would get, and if senators would be able to offer proposals on any provision of immigration or just the four pillars being proposed by President Donald Trump: border security, broadly defined to included physical security and enforcement powers; ending the diversity visa lottery; curtailing family-based migration; and a permanent solution for the Deferred Action for Childhood Arrivals program , the policy that allowed undocumented immigrants brought to the US as children to stay in the country and that Trump has decided to end.

Negotiators hope to have plan Tuesday

Democrats and moderate Republicans negotiating with them, meanwhile, said they have nothing finalized -- yet. But the goal, they said, is to have something by Tuesday.

"We are continuing to really talk turkey, legislative language etc," said Democrat Sen. Tim Kaine, who is part of a bipartisan group negotiating on the topic. "We're making progress, but," he added, trailing off. "It may be (ready Tuesday), not for sure."

"We're close but we're not ready right yet," said Maine Republican Sen. Susan Collins, who co-leads the group.

"No, that's being worked on," Arizona Republican Sen. Jeff Flake said when asked if he was concerned about the lack of an offer. "We'll have that by tomorrow."

What's next

The Senate is stepping into uncharted waters for recent history. Different amendments will be offered that will compete for 60 votes, the threshold to advance legislation in the Senate. If a proposal can reach that number, it will likely pass the upper chamber -- but face an uncertain future in a House and White House that has not made any commitments to the eventual result.

The No. 2 Senate Democrat and longtime immigration reform advocate, Durbin said the only path forward he could see was getting at least 11 Republicans to join the 49 Democrats in the Senate.

"I'd feel much more optimistic and if I knew ... 11 Republican names," Durbin told reporters Monday afternoon. "We believe we have five or six strongly moving in our direction, and I feel that are another five are within reach, and I'm constantly talking to every Republican privately."

On Monday afternoon, senators took to the floor of the upper chamber to speak on the upcoming debate, with a group of Republicans supporting a version of the White House framework that they introduced Sunday night

That proposal got the support of Senate Majority Leader Mitch McConnell, though he did not indicate that he would tip the scales in its favor procedurally.

"It's our best chance to producing a solution that can actually resolve these matters," McConnell said.

Democrats say the White House framework supported only by Republicans couldn't pass.

"The only enemy to this process is overreach," Senate Minority Leader Chuck Schumer said. "Now is not the time nor the place to reform the entire legal immigration system."

Monday afternoon, Durbin said he had faith in McConnell to keep his promise to be neutral in the process.

"I told Mitch McConnell looking him in the eye and said, 'I trust you ... and I defended you among some Democrats who were skeptical,'" Durbin said. "I was skeptical. I'm defending him now -- I think he's going to play it straight."