For many members of Congress, being open to an immigration overhaul is increasingly a matter of political survival: About 27.3 million Latinos are qualified to vote in this year’s election, according to the Pew Research Center, constituting a record 11.9 percent of eligible voters. Latino organizations have been focused on encouraging turnout in a year some speculate could make Hispanic voters an electoral heavyweight for years to come.

In 164 of 435 districts, the share of eligible Latino voters increased by at least one percentage point during the past four years, said David Wasserman, an editor for the nonpartisan Cook Political Report who focuses on House races. Most of the districts that have had the most growth are held by Democrats, though 70 of those 164 are in Republican hands, he said.

Representative Mike Coffman of Colorado, for instance, has seen the emerging power of the Hispanic vote. A Republican who voted in his first term against a bill to create a path to legalization for young people who came to the United States as children, he watched — and reacted — as redistricting abruptly dealt him a constituency in which one in five residents is Latino, compared with less than one in 10 when he was first elected. Now Mr. Coffman is among those House Republicans most engaged in conversations about the need for an immigration overhaul.

Yet for lawmakers like Mr. Dold and Mr. Coffman, it is increasingly difficult to put distance between themselves and Mr. Trump.