WASHINGTON — Even as they were popping corks on Thursday night after a strong bipartisan vote in the Senate to pass an immigration bill, supporters of the overhaul were laying plans for the far more difficult task of moving something similar through the Republican-controlled House.

Speaker John A. Boehner of Ohio insisted on Thursday that the House would not take up the Senate bill and would pass its own measure only if a majority of Republicans backed it, instead of relying more on Democratic votes.

As a sign of the conservative direction of the debate in the House, its Judiciary Committee recently approved a bill to enforce immigration laws away from the nation’s borders that was much tougher than anything from the Senate. The House has yet to produce a bill that includes a core piece of the Senate legislation: a path to citizenship for 11 million unauthorized immigrants in the country.

While supporters are hardly confident, they say House Republicans will soon discover a crucial difference this year from failed immigration efforts of the past. They say their coalition is broader and far more energized and committed than in 2007, when an immigration overhaul by President George W. Bush did not even reach a vote in the Senate.