After months of relative quiet on the subject of immigration reform, President Obama reclaimed center stage in an event in the East Room of the White House Thursday, urging the Republican-controlled House to take up bipartisan legislation passed in June by a big margin (68-32) in the Senate.

“It doesn’t get easier to put off,” he said, a pointed reminder to Republicans that the politics are stacked against them if they punt on an issue of central importance to the fastest growing bloc of voters in the country. Neutralizing the Democrats’ advantage among Hispanics is crucial to the GOP’s presidential prospects, and could improve Congress’ image in the wake of the government shutdown. “Rather than create problems, let’s prove to the American people that Washington can actually solve some problems,” Obama implored.

Among those assembled in the East Room for the president’s remarks was Frank Sharry, founder and director of America’s Voice and a longtime activist for immigration reform.

Asked what he was thinking as he listened to Obama’s 12-minute speech, he termed it “a modest push,” noting that Obama has been “remarkably restrained” on the issue when you consider that overhauling the nation’s broken immigration system is his top second-term priority. Obama sidelined himself in deference to Republicans who needed room to build support without being aligned with a president so many in the GOP caucus reflexively dislike.

But now with the shutdown behind them and Republicans on the defensive, Obama saw an opening to get back in the game. His message, says Sharry: “‘Hey, I’m flexible,’ which after the shutdown politics was important, and he implied ‘if you don’t do it, I’m coming after you.’” For Obama and the Democrats, immigration reform is a win-win issue. They want an overhaul for the country and their constituents. If they don’t get it, they will hammer Republicans in demographically changing districts in California, Nevada, and Florida, where they could likely pick up seats—not enough to win control of the House, but, paired with what Sharry calls “the shutdown narrative,” Democratic operatives are salivating at the prospect of waging that campaign.

Some Republicans understand the stakes, and former vice-presidential candidate and budget maven Paul Ryan is at the center of a newly energized backroom effort to craft legislation that would deal with the thorniest aspect of immigration reform for Republicans: the disposition of 11 million people in the country illegally. Rep. Raul Labrador (R-ID), an early advocate of reform who abandoned the effort some months ago, argues that Obama’s tough bargaining during the shutdown means Republicans can’t trust him on immigration. “When have they ever trusted him?” asks Sharry. “Nobody is asking them to do this for Obama. They should do this for the country and for themselves.... We’re not talking about tax increases or gun violence. This is something the pillars of the Republican coalition are strongly in favor of.”

Among those pillars is Chamber of Commerce President Tom Donahue, who on Monday noted the generally good feelings about immigration reform among disparate groups, among them business and labor. He expressed optimism that the House could pass something, go to conference and resolve differences with the Senate, get a bill and have the president sign it “and guess what, government works! Everybody is looking for something positive to take home.”

The Wall Street Journal reported Thursday that GOP donors are withholding contributions to lawmakers blocking reform, and that Republicans for Immigration Reform, headed by former Bush Cabinet official, Carlos Gutierrez, is running an Internet ad urging action. Next week, evangelical Christians affiliated with the Evangelical Immigration Table will be in Washington to press Congress to act with charity toward people in the country without documentation, treating them as they would Jesus. The law-enforcement community has also stepped forward repeatedly to embrace an overhaul.

House Speaker John Boehner says he wants legislation, but not the “massive” bill that the Senate passed and that Obama supports. The House seems inclined to act—if it acts at all—on a series of smaller bills starting with “Kids Out,” a form of the Dream Act that grants a path to citizenship for young people brought to the U.S. as children; then agriculture-worker and high-tech visas, accompanied by tougher border security. The sticking point is the 11 million people in the country illegally, and finding a compromise between Democrats’ insistence that reform include a path to citizenship, and Republicans’ belief that offering any kind of relief constitutes amnesty and would reward people for breaking the law.

The details matter hugely, but what a handful of Republicans, led by Ryan, appear to be crafting is legalization for most of the 11 million but without any mention of citizenship. It wouldn’t create a new or direct or special path for people who came to the U.S. illegally or overstayed their visa. It would allow them to earn legal status through some yet-to-be-determined steps, and once they get it, they go to the end of a very long line that could have people waiting for decades. The Senate bill contains a 13-year wait.

However daunting that sounds, the potential for meaningful reform is tantalizingly close with Republicans actively engaged in preparing their proposal, pressure building from the business community and religious leaders, and a short window before the end of the year to redeem the reputation of Congress and the Republican Party after a bruising takedown. The pieces are all there for long-sought immigration reform. We could be a few weeks away from an historic House vote, or headed for a midterm election where Republicans once again are on the wrong side of history and demography.