After a presidential election in which Latino voters rewarded President Obama while punishing Republicans for their positions on immigration, Republican leaders and prominent conservatives moved quickly this week to shift to new ground, saying they could support some kind of legislation to fix illegal immigration.

The prospects for an immigration overhaul next year improved with stunning speed after the vote, with John A. Boehner, the speaker of the House, who had long resisted any broad immigration bill, saying on Thursday that “a comprehensive approach is long overdue.” Haley Barbour, a Republican elder statesman and former governor of Mississippi, echoed Mr. Boehner, and Sean Hannity, the conservative talk show host — in a startling turnaround — joined calls for measures opening pathways to legal status for illegal immigrants.

One of every 10 voters who cast ballots on Tuesday was a Latino, and they favored President Obama, with 71 percent of their votes, compared with 27 percent for Mitt Romney, forcing Republican leaders to wonder if they could ever regain the presidency without increasing their appeal to Hispanic Americans.

Mr. Obama wasted no time, renewing in his acceptance speech early Wednesday his promise to move “in the coming weeks and months” on “fixing our immigration system.”