Roy Beck, a founder of NumbersUSA, a group that opposes the Senate legislation, acknowledged in an interview that the push by conservatives — evangelical leaders in particular — worried him somewhat, leading him this week to urge his one million followers nationwide to step up their calls and e-mails to Congress.

“There is the potential this could shift some support,” Mr. Beck said.

Backers of the effort estimated that about 30 House Republicans, like Mr. King and Mr. Yoho, would not support immigration legislation under almost any condition. But they believe they can piece together a majority of the Republican caucus to pass certain bills, moving the debate to a committee of House and Senate negotiators, who could try to agree on a comprehensive package.

The contention centers on what to do with the illegal immigrants already in the country, which Democrats in both chambers say must be addressed in any final deal. Many House Republicans have already expressed support for proposals to strengthen border security and make it easier for high-skilled workers and farm laborers to get visas, all elements of the Senate package.

There have been hints of possible compromise. Representative Mick Mulvaney, Republican of South Carolina, who is one of the House’s more conservative members, said he could support “a path to status” for immigrants who are here illegally.