President Obama’s victory, fueled by massive support among Latino voters, has produced a new dynamic in Washington: a Democratic president who owes a huge debt to Latinos, and a Republican Party desperate to find a way to avoid another landslide loss among one of the fastest-growing sections­ of the electorate.

“Democrats want reform, and Republicans need it. That’s the new dynamic,” said Frank Sharry, the executive director of America’s Voice, an immigrant advocacy group.

Even before the election, ­Republican strategists were increasingly worried that the GOP’s hard-line opposition to immigration reform was endangering its long-term viability by making the party toxic to Latinos.

Since those fears were borne out Tuesday, several Republican leaders, including House Speaker John Boehner, have said the GOP needs to reckon with immigration reform and pass a comprehensive bill.

“This issue has been around far too long,” Boehner said in an interview with ABC News on Thursday. “I’m confident that the president, myself, others, can find the common ground to take care of this issue once and for all.”

But immigration has divided the GOP for years, and some Republican lawmakers are already pushing back, setting up a potential internecine battle if immigration reform comes up next year, as expected.

Most comprehensive proposals that have circulated in the past would have increased border security, instituted verification requirements for employers to ensure they are not hiring illegal workers, changed the visa system to bring in more high-skilled workers, and provided a pathway to citizenship for the estimated 11 million illegal immigrants here now.

Under most comprehensive plans, those immigrants would have to pass a background check and prove they can speak or are studying English, among other requirements.

President George W. Bush proposed a comprehensive plan in 2004 that would have allowed­ illegal immigrants to apply for legal status and stay in the country for up to six years.

But after years of negotiations, the plan, backed by Bush and senior lawmakers in both parties, collapsed in 2007. In the Senate, 15 Democrats and 37 Republicans voted against the proposal; many GOP lawmakers complained that opening any pathway to citizenship for illegal immigrants would reward lawbreakers and punish immigrants who had played by the rules.

Since then, the GOP has adopted an increasingly hard-line tone on immigration, opposing legislation that would provide immigrants who entered or stayed in the United States illegally a way to gain legal status.

But on Sunday, as the immigration issue reemerged with new prominence, Senator Chuck Schumer, a Democrat from New York, said he had resumed negotiations with Republican Lindsey Graham of South Carolina on an immigration overhaul.

Schumer said the two had drawn up a blueprint that he said has ‘‘the real potential for bipartisan support, based on the theory that most Americans are for legal immigration, but very much against illegal immigration.’’

The senators promoted similar overhaul plans in separate interviews on Sunday news shows, and both said no path to citizenship should be available until the country’s borders are secure. Graham did not directly mention working on a plan with Schumer, the chairman of the immigration subcommittee of the Senate Judiciary Committee.

Already, the GOP’s new openness on immigration reform is opening up fissures in the party.

After Boehner’s interview last week, Representative John Fleming, a Louisiana Republican, told the Associated Press that Boehner was ‘‘getting ahead of House Republicans when he commits to getting a ‘comprehensive approach’ to immigration.’’

‘‘There’s been zero discussion of this issue within the conference, and I’m urging the speaker to talk with House Republicans before making pledges on the national news,’’ he said.