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Forcing the vote, using what's known as a discharge petition, would amount to a challenge to leadership. Still, it's one that more and more supporters of immigration reform seem to be comfortable with.

While Denham said he has the numbers for a discharge petition, he’s expressed reluctance to commit to filing the motion.

Still, leadership is dismissing the notion altogether.

"I don't believe discharge petitions are the way to legislate," McCarthy told The Hill. "I don't believe members in the conference believe that, either."

"[Denham's] got 50 co-sponsors, so they would have to convince more than half of those to not sign on to a discharge," he said.

"You should not need to force a vote or have a discharge petition or have a protest out in the front yard when you have 250-plus people that are continuing to sign on," said Denham. "So this growing number, we think, will not only show the Speaker but will show the president that this is what the people's House wants to do."

At issue is legislation to restore protections from the Deferred Action for Childhood Arrivals (DACA) program, which Trump ended in September.

After a flurry of legislative attempts, Congress essentially gave up on DACA after the courts put a stay on Trump's rescission of the program, allowing its 690,000 recipients to keep their benefits: work permits and protection for deportation.

During that legislative whirlwind, four rough proposals to deal with the future of so-called Dreamers — DACA recipients and other immigrants brought to the country illegally as minors — were considered.

Under Denham's rule, those four proposals would be voted on at the same time on the House floor, and whichever idea received the most votes over 218 would be sent to the Senate.

Under "Queen of the Hill," slots are assigned to specific legislators to propose their own bills.

The array of options has not been enough to convince Ryan or McCarthy, who've pledged to support Goodlatte's proposal, that a Queen of the Hill strategy is the path forward.

But for some House conservatives, it's an understandable strategy — if not one they'll sign on to.

"Jeff Denham is representing the 750,000 people he's got the privilege of representing, and he believes he needs to have a vote that deals with DACA," said Meadows.

"I can't criticize that. Even though it may not be my path, it's certainly to be applauded," he added.

Still, Ryan's stuck to his support for the Goodlatte bill and to laying blame on Democrats, who declined in March an offer from the White House that would extend DACA until 2020 in exchange for $25 billion in border security funding.

"Republicans made three good-faith offers during the omnibus negotiations and Democrats declined all of them. We continue to work to find the support for a solution that addresses both border security and DACA," said AshLee Strong, a spokeswoman for Ryan.

That's started to wear at the patience of Republicans who favor immigration legislation, particularly those from Latino-heavy districts or suburban districts where a DACA solution has polled well.

Asked if a discharge petition was the next step, Newhouse said, "I'm getting close to the point where I think we have to get this done and Congress has to work this through."

"This is almost a last-ditch effort, which we may be getting close to that, but I'm not sure we are yet. But if that's an option to have it available, I'm OK with that," he added.

Newhouse has not signed on to Denham's resolution, although he supports it, because he is a member of the Rules Committee.

"I don't think that we should operate like that, in terms of what we think the president may or may not do," said Taylor.

"I know that [Ryan] said that before. Respectfully, I'm not sure if I'd agree with that," he added.