WASHINGTON -- Immigration reform, one of the most complex and divisive issues facing the country, suddenly has new life on Capitol Hill.

President Barack Obama vows to make it a priority early next year, and Republicans, chastened by the November election’s outcome, are determined to put their imprint on long-elusive reforms.

But there is significant disagreement over how to proceed and how far to go, a divide captured by two Floridians, Rep. Mario Diaz-Balart and Sen. Marco Rubio, Cuban-American Republicans from Miami who could be crucial bridges to a bipartisan solution.

Rubio, lifted but also constrained by the issue due to his status as a conservative star with presidential ambitions, wants to go step by step.

“Usually when Congress approaches issues of this magnitude with one big bill, it almost always requires you to swallow five really bad ideas in exchange for one good one,” he said in an interview. “We’re going to get better public policy if we focus on each of these apart from each other.”

Diaz-Balart is echoing Democratic calls, and emerging support among other Republicans, for a comprehensive approach that includes everything from better enforcement to handling the 11 million undocumented residents already in the United States.

“To pretend you can fix a broken system by tweaking one aspect is not solving the problem. It’s broken from A to Z,” Diaz-Balart said.

“That’s the million-dollar question, which is the better approach?” said Jennifer Korn, executive director of the Hispanic Leadership Network, who was part of President George W. Bush’s administration when sweeping reform collapsed in 2007 under opposition from both parties.

Doing it Rubio’s way could bring on enough Republicans to achieve a deal. But that approach also provides political cover and could leave aside other important policy changes.

“I know people that are in the immigration advocacy community are concerned that we’ll only pass the easy stuff and leave the hard stuff. I don’t want to see that, either,” Rubio said. “We can figure out a sequence.”

The Diaz-Balart approach could repeat past failure.

“They both feel a growing and unavoidable sense of responsibility as Republican Hispanics to lead on this issue,” said Ana Navarro, a Republican strategist who is close to both Florida lawmakers. “It’s an opportunity for both of them to burnish some legislative credentials.

“Marco has seen that comprehensive reform has failed and is trying to think outside the box,” she said. Still, “there’s no point to bringing in more Republicans if you lose Democrats. An incremental approach would be a very hard sell.”

For Republicans, the 2007 effort marked the rise of a conservative backlash against anything deemed to be amnesty and forced politicians, including Rubio during his 2010 Senate run, to adopt harder-line positions on enforcement. That continued in the 2012 election, with the sentiment captured by Mitt Romney, who said he would make things so difficult for undocumented residents that they would “self-deport.” Romney got a dismal 27 percent of the fast-growing Hispanic vote.

Now Rubio, 41, is trying to manage his party’s newfound eagerness to win over Hispanics as he feels out a possible run for president. So he speaks broadly about the issue, or focuses on changing the tone with which Republicans address it.