But any Republican efforts to play gradual, political small ball with immigration may be stymied by the president’s strategy: Obama is inclined to push for one big bill that includes the one thing the Rubio-Ryan axis might want to avoid — a pathway to citizenship for undocumented workers now in the country.

Advisers say Obama plans to begin a public campaign shortly after the fiscal cliff is resolved, using social media and grass-roots activity to harness business groups, liberal nonprofits, and the activists who helped generate a record Hispanic turnout in November. The White House is calculating that smart Republicans will put their stamp on the legislation, broadening support. “The fact that we want to move forward on immigration reform doesn’t mean we can write the bill, whole cloth,” a presidential adviser said.

The issue also carries risk for Obama. Labor unions are leery of immigration reform because of the potential competition posed by more legal workers. AFL-CIO President Richard Trumka has sounded conciliatory, arguing that the labor movement was founded by immigrants. But unions will fight fiercely over the exact numbers of immigrants who might benefit. “You also can’t discount the issue of race, whether consciously or subconsciously,” one labor official said.

However the fight unfolds, this will be a real day of reckoning for Republicans. Prior to November, they appeared to be in denial about the steady decline of the white voter — and the rise of Hispanics. Every four years, the percentage of the electorate that is white declines — this time to a historic low of 72 percent, down from 87 percent in Bill Clinton’s 1992 win. At the same time, Hispanics surged, from 3 percent in Clinton’s first election to 10 percent last month.