The leader of the House Freedom Caucus warned on Wednesday that Speaker Paul Ryan Paul Davis RyanKenosha will be a good bellwether in 2020 At indoor rally, Pence says election runs through Wisconsin Juan Williams: Breaking down the debates MORE's (R-Wis.) job could be in jeopardy if he doesn't handle the debate over immigration to conservatives' liking.

The Senate is attempting to conduct an open-ended floor debate on legislation that would both allow young undocumented immigrants brought to the U.S. as children to stay in the country and enhance border security, although leaders have so far been unable to agree on which proposals should get votes.

Rep. Mark Meadows Mark Randall MeadowsTrump reacts to Ginsburg's death: 'An amazing woman who led an amazing life' Trump carries on with rally, unaware of Ginsburg's death United Airlines, unions call for six-month extension of government aid MORE (R-N.C.), the Freedom Caucus chairman, expressed frustration that the House wasn't moving as effectively as the Senate. And members of the roughly 40-member conservative group are wary of more moderate proposals being floated in the upper chamber, some of which would go as far as giving the young immigrants a path to citizenship.

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"It is the defining moment for this Speaker. If he gets it wrong, it will have consequences for him. But it will also have consequences for the rest of the Republican Party," Meadows said at an event on Capitol Hill moderated by the Heritage Foundation.

"I don’t think there’s really conversations right now that would involve a new Speaker. But there are certainly conversations that would involve new leadership," Meadows added.

Conservatives are pushing for legislation authored by House Judiciary Committee Chairman Bob Goodlatte Robert (Bob) William GoodlatteNo documents? Hoping for legalization? Be wary of Joe Biden Press: Trump's final presidential pardon: himself USCIS chief Cuccinelli blames Paul Ryan for immigration inaction MORE (R-Va.) that establishes new restrictions on legal immigration in exchange for establishing legal status for roughly 700,000 young immigrants who were permitted to work and go to school by the Obama-era Deferred Action for Childhood Arrivals (DACA) program.

It would authorize funds for President Trump Donald John TrumpObama calls on Senate not to fill Ginsburg's vacancy until after election Planned Parenthood: 'The fate of our rights' depends on Ginsburg replacement Progressive group to spend M in ad campaign on Supreme Court vacancy MORE's promised wall along the Mexican border, crack down on sanctuary cities, end family-based migration and eliminate the diversity visa lottery program.

House GOP leaders are whipping the Goodlatte bill this week to see if it has enough votes to pass.

Meadows said that GOP leaders should move more quickly on an immigration bill, especially given that the slower-moving Senate has already started debate.

"Anytime that the Senate can take up an immigration bill quicker than the House is a day that should defy history. I mean, the Senate is notorious for taking naps and not taking votes," Meadows said.

Trump has proposed granting a path to citizenship for nearly 2 million young immigrants brought to the U.S. as children — which would go beyond the DACA population — in exchange for restricting family migration, authorizing funds for his border wall and ending the diversity visa lottery.