President Obama is moving full speed ahead on a broad immigration proposal that Congress will debate in the coming months. The plan from the White House will be comprehensive, seeking a path to citizenship for 11 million undocumented immigrants, creating new temporary-visa programs for low-skilled foreign workers when American workers aren’t available, mandating electronic verification of all new hires, and giving more green cards to foreign math and science graduates at U.S. universities.

Republicans are wary of Obama’s approach, saying they would rather take up individual pieces of the package separately rather than grapple with a complicated legislative monster. House Republicans are expected to take a less ambitious track on immigration with a series of smaller bills that start in the Judiciary Committee, which is led by conservative Rep. Bob Goodlatte of Virginia.

Obama’s larger-scale strategy plays into the fears of Republicans (and some immigration-reform advocates) that he cares more about the politics of immigration than completing actual legislation. Republicans are well aware that the stance of their presidential nominee, Mitt Romney, cost them with voters. The Massachusetts governor’s suggestion that illegal immigrants should “self deport” morphed into both a joke and an insult. Why would Obama give away that advantage by signing something into law that Republicans could claim as a victory? But if he insists on a broad package and it fails, he can easily blame the other party. It would be a win-win for Obama and Democrats, but not for undocumented immigrants or employers who are desperate for workers.

The hand-wringing is legitimate, but it’s probably overstated. All signs indicate that Obama is serious about pushing for an immigration overhaul. He mentioned it several times—unprompted—in a news conference Monday that was otherwise devoted to gun control and debt-ceiling negotiations. People close to the administration say that Obama has not backed down from his commitment to immigration, even though the high-profile debates on gun measures and the debt ceiling could easily serve as distractions.

“Washington should be able to walk and chew gum at the same time,” said Los Angeles Mayor Antonio Villaraigosa at the National Press Club on Monday. Villaraigosa is a leader on immigration and is also on several lists of possible Cabinet nominees for Obama’s second term.

The president’s commitment to immigration is a huge step toward finishing legislation this year. Using the bully pulpit, Obama might be able to cajole and/or shame some Republicans into voting for a package if it includes some their priorities, such as visas for high-skilled foreign workers and entrepreneurs. Business leaders, including U.S. Chamber of Commerce President Thomas Donohue, already are planning their own outreach to encourage Republicans to vote for the legislation.

It will be tricky, but not impossible to pass the package in Congress. The Senate will act first, with House members watching closely to see what works and what doesn’t in the debate. It’s not out of the realm of possibilities that the Senate could pass a comprehensive immigration bill. It did in 2006, with the support of 23 Republicans. Senate Majority Leader Harry Reid is already planning floor debate on a broad bill sometime early this year. Several Senate Republicans—Arizona Sens. John McCain and Jeff Flake, Utah Sen. Mike Lee, and South Carolina Sen. Lindsey Graham—are negotiating with Democrats on the details of the legislation.

In 2006, the Senate-passed bill died in the House, but House leaders are in a different position this time around. Back then, Speaker Dennis Hastert produced the winning argument against the immigration bill—shore up the border first. Now, the administration has answered that call, boasting the most aggressive deportation policies ever. Illegal border crossings have fallen by about two-thirds since 2006.

Republicans also are well aware that Obama got three-quarters of the Hispanic vote in the last election, which means they need to shift on immigration or cast off any hope of winning over Latino voters. House Speaker John Boehner wants to take the issue off the table for Republicans, although he has had difficulty stemming rebellion in his own party.

Immigration could expose ugly fights inside the Republican Party, but there are options for pushing an Obama-supported bill through the House. One option would be for the House to pass an immigration bill that includes only some components of a broader package—electronic verification of new hires and high-skilled visas, for example—and use that bill as a vehicle for a House/Senate conference committee. That’s how the transportation legislation finally passed last year.

Another option would be for Boehner to waive his usual rule that a majority of Republicans must support a bill before he puts it on the floor. He did that late last year to avoid tax increases that would have gone into effect automatically without the House’s support. It was a bold move, but Boehner recognized the damage that would be done to his own party politically if he allowed tax rates to go up. With a similar threat of political damage, it’s possible that scenario could play out again.