In all, 84 percent of House Republicans represent districts that are 20 percent or less Hispanic.

Of course, Republicans without a large bloc of Hispanic constituents could still back changes to immigration law, and vice versa. But if Speaker John A. Boehner abides by the Hastert rule — which says that a bill should only be brought to a vote if the majority of the majority supports it — then House legislation overhauling the nation’s immigration system will have to rely on a substantial number of Republicans who represent mostly white districts.

If any legislation cannot clear that hurdle and win a majority of the majority, its best hope may be to follow the path of the Hurricane Sandy relief bill and the vote on the deal on the so-called fiscal cliff. In both cases, Mr. Boehner allowed a vote without a majority of his caucus supporting the legislation. Both bills passed with a majority of Democratic votes and supported by a minority of Republicans.

Which Republicans might feel compelled to back an immigration overhaul? One place to look would be the Republican-held districts with the largest Hispanic communities.