Washington (CNN) For veterans of immigration reform, it's déja vu all over again. And it could spell another disappointment for lawmakers who have long sought a compromise on the issue.

In the wake of the government shutdown, which Democrats in the Senate agreed to end in exchange for a vague commitment to debate immigration on the Senate floor, reality is dawning that the House is taking a much different approach — and neither party in either chamber has figured out a plan to reconcile the differences.

It's leaving lawmakers and staff feeling the echoes of 2013, when the Senate passed a bipartisan immigration bill that died when the House did not take up that bill or any other that would be similar. Vermont Sen. Pat Leahy, who called passing that Senate bill one of his "proudest moments," said it died in the House because of an informal rule against bringing legislation without the support of a majority of the Republican conference, and it just might again for the same reason.

"Which must have given Speaker Hastert some pleasure, probably, sitting in his prison cell serving his sentence for ( charges related to covering up allegations of ) child molestation, to see they're still following the sacred Dennis Hastert Rule," Leahy said. "You've got to have members in both parties who are more interested in substance than soundbites."

Signs of Trouble

Hours after Senate Majority Leader Mitch McConnell announced on the floor Monday that he intended to hold an open debate on immigration in the Senate, even if no broad agreement is reached by the time government funding runs out, the House majority whip poured cold water on the notion that his chamber would follow suit.

Rep. Steve Scalise, a Louisiana Republican, told reporters Monday afternoon that McConnell's pledge on the Senate floor to turn to immigration in February held little weight on the House side.

"There were no commitments made in the House," Scalise said.

In fact, even as a bipartisan group of senators is pushing McConnell to find a bipartisan compromise that can pass the Senate, where Republicans hold only hold a 51-49 majority and 60 votes are needed to advance legislation, Republicans on the House side are pushing their leadership to seek as conservative a bill as possible.

The Republican Study Committee, a group of more than 150 House Republicans, on Tuesday announced it would back a hardline immigration bill that has a rough path to pass the House, let alone the Senate. The move follows efforts by the House Freedom Caucus, a smaller group of vocal conservatives, that extracted a promise from leadership to whip the bill in exchange for their votes on government funding.

"Do I empathize with (leadership)? I do," said RSC Chairman Rep. Mark Walker of North Carolina in an interview about the decision Monday. "You have so many factions in the House ... so you've got a lot to wrestle with. At the same time, when you have a bill like this that has the support of a majority of the conference, I believe this is the foundational piece to move forward."

Scalise said that any bill that passed the House would need to include funding for a border wall and could not include "amnesty." But Scalise wasn't clear on how he defined that and whether it would mean a pathway to citizenship for recipients of the Deferred Action for Childhood Arrivals policy , a program that protected young undocumented immigrants who came to the US as children that Trump has decided to end. A version that could pass the upper chamber would almost certainly require a pathway to citizenship.

"Ultimately, we've got to see how all sides can come together," Scalise said. "Let's see if the Senate can come together with something that President Trump can support. And I think there's a deal to be made, but in my mind it would not include amnesty and has to include border security and funding of a wall."

Freedom Caucus Chairman Rep. Mark Meadows of North Carolina told CNN on Monday that the House should pass something "as conservative as it possibly can be" and then go to conference with whatever the Senate passes, but he said the bill "can't start in the Senate."

Paths forward

The disconnect has the potential to lead to an impasse that can't be breached without the President's firm support of a path forward.

One veteran House Democratic aide struck a pessimistic note about the situation, especially after the failed attempt in the Senate to push the issue forward through the shutdown tactics.

"It is really hard to see a way out of it right now. I'm hopeful, still, but," the aide said, trailing off. "It's not strategically bad to go Senate first, but it's bad when that's your only strategy and you don't have a House strategy other than, 'Well, we'll magically get the Senate bill through or the House will feel forced to do it.'"

A senior House GOP aide expressed frustration that the Senate side was taking the same approach as in 2013.

"It's the same mistake they've made every single time," the aide said of the Senate's plan. "It's like Groundhogs Day. Somehow, include the House."

But that seemed to be the hope, if the President could be engaged on it.

"Get a big vote in the Senate and have the President support it, I think that's it," said Arizona GOP Sen. Jeff Flake, who has pushed for a compromise, when asked by CNN on Tuesday how to get a bill through the House.

Oklahoma Republican Sen. James Lankford, who is working on immigration in the Senate now but was in the House during the last effort, said the path will require White House leadership:

"The best thing that could happen is the White House put out a proposal and then try to work with House and Senate Republicans and say this is where our boundaries are and then try move from there."

The worst plan, said Florida Republican Rep. Mario Diaz-Balart on CNN on Tuesday, would be to work in isolation.

"The concept that either a House bill can be shoved through the Senate or a Senate bill can be shoved through the House just doesn't tend to work," Diaz-Balart said. "It has to be bipartisan, with buy-in from the White House, otherwise there is nothing doing, and bicameral."