Much attention has been paid to the handful of seats in New York, New Jersey and California that are represented by Republicans but voted for Hillary Clinton last year. But even with district lines drawn to favor Republicans in many states, the swelling antipathy toward Mr. Trump threatens to breach the party’s defenses and stretch the congressional battlefield beyond the dimensions Republicans and Democrats anticipated a year ago.

“There’s no illusion about the storm that’s coming,” said Representative Tom Cole, Republican of Oklahoma, invoking last month’s governor’s races and last week’s Senate special election. “If you had any doubts, they were wiped away after New Jersey, Virginia and Alabama.”

From Texas to Illinois, Kansas to Kentucky, there are Republican districts filled with college-educated, affluent voters who appear to be abandoning their usually conservative leanings and newly invigorated Democrats, some of them nonwhite, who are eager to use the midterms to take out their anger on Mr. Trump.

“If you look at the patterns of where gains are being made and who is creating the foundation for those gains, it’s the same: An energized Democratic base is linking arms with disaffected suburban voters,” said Mayor Rahm Emanuel of Chicago, who as a member of Congress in 2006 helped Democrats win back the House. “The president’s conduct has basically given voters this permission slip to go against the Republicans.”

Congressional Republicans are scrambling to fortify their defenses.

On Wednesday, the last five leaders of the House Republican campaign arm privately addressed Republican lawmakers, outlining the sort of suburban districts most at risk and imploring members to contribute to their colleagues. The former Representative Thomas M. Reynolds of New York said it had been aimed at dozens of lawmakers elected since 2010 who had never faced a Democratic wave.