A growing number of House GOP conservatives are pressuring Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellObama 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 (R-Ky.) on Thursday to invoke the "nuclear option" and change the chamber's rules to pass a bill defunding President Obama's executive actions on immigration.

Reps. Raúl Labrador (R-Idaho) and Tim Huelskamp Timothy (Tim) Alan HuelskampDemocratic-linked group runs ads in Kansas GOP Senate primary Cure for cancer would become more likely if FDA streamlined the drug approval process Emails show climate change skeptics tout ‘winning’ under Trump MORE (R-Kan.) said McConnell should change Senate rules, so the House-passed Department of Homeland Security (DHS) funding bill, which includes language to revoke Obama's immigration-related actions, can bypass a Democratic filibuster in the upper chamber.

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Rep. Mick Mulvaney Mick MulvaneyMick Mulvaney to start hedge fund Fauci says positive White House task force reports don't always match what he hears on the ground Bottom line MORE (R-S.C.) also endorsed the idea at a Thursday news conference. He said there’s a “way to change the rules to allow us to move forward” and “take away the ability to filibuster.”

Rep. Mo Brooks (R-Ala.) was the first House Republican to advocate such a rules change Wednesday evening, arguing that now-Senate Minority Leader Harry Reid (D-Nev.) had established a precedent during his time in the majority.

Republican senators, however, immediately sought to quash the idea.

“The answer is not to change Senate rules,” Sen. Ted Cruz (R-Texas) said during the same news conference at which Mulvaney spoke. "The answer is for Senate Democrats not to be obstructionists.”

“I don’t think that’s an option we’re looking at right now,” freshman Sen. Dan Sullivan (R-Alaska) added, arguing that things should move forward according to current Senate rules.

Labrador and Huelskamp said at an event held in conjunction with the Heritage Foundation that the executive actions merited something as dramatic as a change to the Senate's long-standing rules. They argue that spending bills should only need 51 votes, instead of 60, to advance.

"Mitch McConnell can change the rules of the Senate. And this is important enough for Mitch McConnell to change the rules of the Senate," Labrador said.

Huelskamp agreed.

"I don't think Mitch McConnell should let the Senate rules trump the Constitution," the Kansan said.

Labrador suggested Senate Republicans should just "pack up" and go home if they "don't want to fight" on the DHS funding issue.

"If they don't want to fight, if they don't want to work, if they don't want to do the hard work that is necessary to do the will of the American people, then maybe they just need to pack up, and they need to decide that for the next two years, we're just not going to do anything in the Senate," Labrador said.

Otherwise, Labrador argued, Senate Republicans might as well hand the majority back over to the Democrats.

"If we're going to allow seven Democratic senators to decide what the agenda is of the House Republican Conference, of the Senate Republican majority, then we might as well just give them the chairmanships, give them the leadership of the Senate," he asserted.

Brooks first made the suggestion Wednesday night during a floor speech in the House. His basic argument was that McConnell could follow the path already walked by Reid, who in 2013 eliminated filibusters for most executive nominees.

"There's another option. Let's think back for a moment. And let's look at Harry Reid, when he was Senate majority leader and the power that he wielded," Brooks said. "He said, 'I'm not going to let the filibuster stop me from achieving my political goals.'

"Well if Harry Reid and the Democrats can do that, if they can stand up for their beliefs, however wrong those beliefs may be, then where is our Republican Senate leadership? And why aren't they doing the same thing?" Brooks said.

The comments from House conservatives reflect the simmering intraparty tensions between House and Senate Republicans.

When asked by a reporter to respond to Sen. Mark Kirk's (R-Ill.) call that Congress pass a "clean" DHS funding bill with no immigration-related language attached, Labrador didn't mince words.

"Did he tell you how he was going to deal with the immigration issue separately?" Labrador asked. The reporter shook her head. "Then, he has no credibility."

This story was updated at 1:45 p.m. Rebecca Shabad contributed.