A rump caucus of renegade Republicans, backed by well-heeled outside conservative groups, are pushing to change internal House GOP rules to prohibit the party leadership from bringing legislation to the floor that does not have "majority-of-the-majority" support.

The goal? It’s a long shot they can formally codify the so-called “Hastert rule”, named after the former Illinois House Speaker. Rather, the aim of conservatives is to send a clear warning to House Speaker John Boehner (Ohio) not to push legislation this summer and fall on immigration and budget matters unless the measures have popular support among House Republicans.

Former House Speaker Dennis Hastert

Conservative lawmakers – led by freshman Rep. Matt Salmon (R-Ariz) -- began circulating a petition more than a month ago, hoping to secure the 50 signatures required to guarantee a vote in a special Republican Conference meeting on the matter. They declined to say last week how many supporters they have to date -- which means they almost certainly don't have enough.

If the measure is ever put to a vote, it would more clearly show just how many House Republicans are willing to break apart the system -- even at the expense of their own party's leadership.

Boehner has publicly brushed off the effort and GOP aides privately suggest that they don’t think a majority of House Republicans want to further hamstring a leadership team that has already struggled to lasso its caucus. (See "Plan B".) Asked about ]the “Hastert rule”, Boehner said it was unlikely that legislation without significant GOP support would move in the House any time soon.

“I don't intend to bring an immigration bill to the floor that violates what I and what my members of my party, what our principles are,” Boehner told reporters late last week. “And so I continue to believe that you'll see strong bipartisan majorities for bills that we bring to the floor.”

To conservatives, that statement leaves Boehner a lot of wiggle room. They are still stung by the fiscal cliff vote on New Year’s Day, when less than 40 percent of House Republicans supported the tax legislation, and the relief bill for states ravaged by Hurricane Sandy, which passed in mid-January with fewer than 50 Republican votes.

Both bills passed because of near universal support from House Democrats, a scenario that played out in the ensuing months on the Violence Against Women’s Act and more low-profile bill allowing for the purchase of historic battlefield lands.

After a bitter 2012 defeat in the presidential race and a failure to regain the Senate majority, conservatives have come to see the House as their only bastion of power in Washington. The Senate is considering immigration legislation now that includes a pathway to citizenship for millions of illegal immigrants, and some outside activists want the House GOP to block that even as many establishment Republicans argue the party needs to get past the controversial debate in hopes of courting Hispanics in future elections.

“We are writing you today to encourage you to boldly use your majority not only to present a positive conservative vision, but also as the last backstop against the worst excesses of liberalism and Washington deal-making," wrote a coalition of conservative groups in a letter to Boehner recently.  "Liberal Democrats control the White House and the Senate.  We should not help their cause by handing them the keys to the House as well."

Republican aides said that Salmon’s provision would set up a system that would allow an internal vote before the consideration of legislation if at least 25 Republicans signed a letter requesting one. Legislation that failed to garner majority support in that internal GOP conference vote would not be allowed on the floor.

“Codifying the Hastert Rule reinforces our resolve to consider legislation that doesn’t grow government and doesn’t cede legislative power to the minority party," Salmon said in a statement. "I believe this will actually strengthen the hands of our Republican leadership by fostering a unified voice among our conference."

Many Republicans privately suggest that the ranks of conservative antagonists are smaller than it often appears, and that Salmon’s effort is not likely to succeed. They concede that up to 20 Republicans will oppose almost any compromise -- on anything -- as insufficiently conservative, forming the backbone of the effort to have a vote on the Hastert rule. In January, when some of the conservative renegades plotted to deny Boehner the speakership, 16 Republicans initially opposed Boehner on the first ballot, including seven who held back and did not vote at all. (Four of those seven did eventually vote for him after it was clear he had the majority).

After that core group, there are several dozen Republicans who are quite conservative but tend to shy away from open conflict with Boehner and House Majority Leader Eric I. Cantor (R-Va.). Outside of those two circles are another few dozen Members sometimes describe as the "Vote-No-Hope-Yes" Caucus -- those that want to see the party govern effectively but live in fear of facing a primary challenger.

If a vote on imposing the Hastert rule were public, it might have a better chance of passing because many Republicans would fear that the Club for Growth and other conservative groups might support a primary challenger if they opposed it. In a secret ballot, many Republicans would likely see it as a vote of confidence in Boehner and the entire leadership roster – which, if they won on a larger than expected margin, could have the odd effect of actually strengthening Boehner’s hand heading into what is going to be a very difficult summer and fall for the Speaker.