Correction: In a previous version of this story, it was incorrectly reported that Rep. F. James Sensenbrenner Jr.’s immigration bill was defeated in the House in 2006. It was passed by the House in December 2005 but defeated in the Senate in 2006. This version has been corrected.

People shout out against the SAFE Act (Strengthen and Fortify Enforcement Act) Tuesday in the hall outside the House Judiciary Committee hearing on Capitol Hill. (Carolyn Kaster/AP)

The fate of the sprawling immigration reform proposal winding its way through Congress may now be in the hands of some of the most conservative members of the Republican Party.

House Speaker John A. Boehner (R-Ohio) said Tuesday that he will not advance any bill that did not have the support of a majority of the House GOP, which will mean engaging some of the proposal’s biggest detractors and harshest critics.

“I have no intention of putting a bill on the floor that will violate the principles of our majority and divide our conference,” Boehner told his party colleagues in a closed meeting Tuesday, according to GOP aides present. “One of our principles is border security. I have no intention of putting a bill on the floor that the people in this room do not believe secures our borders. It’s not gonna happen.”

In comments to reporters after the meeting, Boehner added that “I don’t see any way of bringing an immigration bill to the floor that doesn’t have a majority support of Republicans.”

That tricky political calculation is one that Boehner has faced repeatedly in the past two years in trying to pass controversial legislation on issue after issue. The approach has often led to failure, as a solid bloc of conservatives has often emerged to derail whatever compromise Boehner and his leadership team managed to work out. But it also required Boehner at the start of the year to depend on Democratic votes to approve federal assistance to communities ravaged by Hurricane Sandy and to adopt the “fiscal cliff” deal.

Despite his closed-door comments, Boehner didn’t rule out relying on Democrats to pass a final version of immigration legislation that could be negotiated between the House and the Senate in the coming weeks, telling reporters Tuesday: “We’ll see when we get there.”

The speaker’s sharpened resolve and his decision to declare it came at a critical moment in the months-long debate over the issue, as the Senate began voting on amendments to its comprehensive bipartisan proposal and bipartisan talks in the House remained at a stalemate over several lingering concerns.

Boehner also faces a sensitive meeting Wednesday with members of the Congressional Hispanic Caucus, who are pushing aggressively for Congress to finally enact changes to the nation’s immigration laws.

Boehner’s comments came at the weekly closed-door meeting of House Republicans, held Tuesday at Republican National Committee headquarters around the corner from the Capitol. He spoke as some of the most conservative Republican lawmakers and well-financed conservative groups are seeking to change internal House GOP rules by blocking legislation from the House floor that does not have “majority-of-the-majority” support.

Boehner admitted that the “Hastert rule” — named after former House speaker Dennis J. Hastert (R-Ill.) — was violated this year when he had to rely on Democratic votes to approve the fiscal-cliff spending deal and federal funding for communities ravaged by Hurricane Sandy, according to senior GOP aides who attended the meeting.

On immigration, however, Boehner said that Republicans have “plenty of leverage” and will not need to rely on Democratic votes to pass a bill.

Aides said privately that Boehner needed to toughen his resolve with colleagues after telling reporters last week that he expected to continue earning “strong bipartisan majorities” for bills brought to the floor.

Boehner acknowledged the tensions with a smile when a reporter asked him whether Rep. Dana Rohrabacher (R-Calif.) was accurate by suggesting this week that the speaker would lose his job if he brought an immigration bill to the floor with insufficient Republican support.

“Maybe,” he said as members of the House GOP leadership team standing behind him laughed.

Boehner’s comments further angered Democrats already upset with Republicans for beginning debate on a controversial immigration proposal to give state and local law enforcement authorities the power to enforce immigration violations.

Rep. Luis V. Gutierrez (D-Ill.), who is leading negotiations with House Republicans on a bipartisan immigration proposal, said that “when it comes to comprehensive immigration reform, we need to be Americans first and party loyalists second.”

Guitierrez and other members of the Congressional Hispanic Caucus plan to attend a long-scheduled meeting Wednesday with Boehner to discuss several issues of concern, chiefly immigration.

They are expected to focus on the GOP-backed bill, which the House Judiciary Committee began debating Tuesday, that would give state and local law enforcement authorities more power to arrest and charge immigrants for over-staying visas or for entering the United States illegally.

The SAFE Act (Strengthen and Fortify Enforcement Act) also would make it a federal misdemeanor to be “unlawfully present” in the United States and would permit state and local governments to draft their own immigration laws, so long as they are consistent with federal statutes. Finally, local law enforcement agencies would be eligible for new federal grants to enforce the immigration laws.

The House passed a similar proposal by Rep. F. James Sensenbrenner Jr. (R-Wis.) in 2005 when it tried to overhaul the nation’s immigration laws. The measure was defeated by the Senate in 2006.

Judiciary Committee Chairman Bob Goodlatte (R-Va.), called the new bill “a game-changing piece of legislation” that will bolster attempts to track illegal immigrants far beyond the borders.

But the committee’s ranking Democrat, Rep. John Conyers Jr. (Mich.), described the proposal as “extreme and heinous,” and said the changes would deter crime victims and eyewitnesses from reporting crimes or providing assistance to local police if they fear potential deportation.

Gutierrez said the bill — coupled with Boehner’s insistence on passing a bill with majority of Republican support — would blunt the GOP’s efforts to attract Hispanic voters.

“You cannot honestly, openly and successfully garner the support of a community and to reach our goal of comprehensive immigration reform when you begin by saying that the 11 million [illegal immigrants] are somehow rapists, murderers, drug dealers, gangbangers and people who do harm,” Gutierrez said.

But Rep. Trey Gowdy (R-S.C.), the bill’s chief sponsor, said his is just one of several bills that will be debated as the committee seeks “a remedy” to the nation’s broken immigration system “step by step and increment by increment.”

A final committee vote on the bill is scheduled for Wednesday.