Members of the Congressional Hispanic Caucus emerged from a meeting with Speaker John A. Boehner on Wednesday afternoon sticking to their hope that the House could pass a comprehensive immigration bill.

But Boehner’s statement Tuesday that he would only bring a bill to the floor that has the support of the majority of the Republican Conference hovered over the meeting. And despite the CHC members’ expression of optimism, it was clear the path to a comprehensive House bill had become more delicate.

“I felt that it was a frank discussion that needed to be had,” said Rep. Raúl M. Grijalva, D-Ariz., a member of the CHC. “He reiterated that this is something that is going to be a job ahead.”

Asked whether the Ohio Republican’s insistence on sticking to the “Hastert rule” made that job more difficult, Grijalva answered bluntly: “Yes.”

“Like we told the speaker, we’re willing to help, we’re willing to continue to talk, but the path to legalization, to us, is essential,” he said. “We’ve compromised a lot on security. We’ve compromised a lot on length of time. We’ve compromised on other areas in there, and it’s getting to the point where you go from, you know, ensuring security to a spirit of meanness, and we’re not gonna go there.”

Grijalva pointed to a recent House floor vote on an amendment offered by Rep. Steve King, R-Iowa, as an example of the roadblocks ahead for securing Democratic interests in a comprehensive immigration overhaul.

King’s amendment to the fiscal 2014 Homeland Security appropriations bill, which passed, would block implementation of a 2012 Obama administration policy halting deportation of young undocumented immigrants brought to the country as children.

The CHC, in a statement issued after the meeting with the speaker, echoed Grijalva’s comments: It made clear it would not accept anything that did not grant “a pathway to earned citizenship that is tough but fair” to most of the 11 million undocumented immigrants in the country.

“It’s going to have to include very strong border security to bring some of their people in, just as it’s going to have to include on our side the reasonable understanding that 11 million or more of the men and women who produce most of the food we consume and do most of the hard labor in this nation are part of the solution,” said Rep. Joe Garcia, D-Fla., a member of the CHC.

Rep. Luis V. Gutierrez, D-Ill., left the meeting resigned to the Hastert rule.

“I left that meeting understanding that there needs to be a majority of Republicans and a majority of Democrats,” the longtime champion of an immigration overhaul said. “We need to come together to do that so that the will of the House of Representatives can be done.”

A bipartisan group of House members has been working for months on a comprehensive bill granting a path to citizenship for many of the 11 million people living in the country illegally while also tightening border security. Members of the group say they have wrapped up their work but have yet to introduce it.

Boehner has backed the group’s effort but has not said he would support the bill. He and other House leaders have said, though, they were committed to regular order, meaning any legislation would first go through committee.

Incremental immigration bills sponsored by House Judiciary Committee Republicans are already moving through markup, despite opposition from Democrats. Whether the bipartisan comprehensive bill will make it through a similar process remains to be seen.

“If we can pass it through committee and if we get it moving, then I think it will go to the floor,” said Rep. Mario Diaz-Balart, R-Fla., a member of the working group. “And if it doesn’t, it doesn’t. The challenge is can we get the votes.”

Getting a majority of House Republicans to back such a bill has always been a long shot. But immigration advocates had hoped to pressure Boehner into putting the legislation on the floor and passing it with the support of a majority of Democrats and a few more moderate Republicans.

If the House comprehensive bill fell apart, advocates were hoping to force Boehner to bring up a similar Senate bill (S 744), now pending on the floor in that chamber.

Boehner’s embrace of the Hastert rule appears to close that alternative. And if he were to reconsider his approach, he’s sure to face a revolt that could threaten his political future. Some of GOP leadership’s most vocal antagonists, among them King, are taking Boehner at his word and will hold him accountable if he should re-evaluate his strategy. Indeed, Boehner already faces an effort from members of the rank and file who want to force a vote in the House Republican Conference to formally codify the Hastert rule.

His allies on Wednesday were unwilling to weigh in on whether Boehner might have gone too far.

“It’s the toughest job around,” said Pat Tiberi, R-Ohio. “I would never question the Speaker.”