NumbersUSA and FAIR have called for the mandatory electronic employment verification, whereby employers would check the immigration status of all hires through a federal system. They also seek a national entry/exit system at every port of entry and border crossing to better track visa overstays.
Both measures are included in the initial Senate proposal.
"All employers have to verify the status of their workers. Period. That is the only way to prevent the next inflow of illegal aliens," said Jenks. "Until you do that, you can't talk about amnesty for those who are here."
As part of its framework for an immigration overhaul, the White House has proposed mandatory electronic employment verification that would be phased in over several years. An official said that the administration is also open to an entry-exit system, although it is not part of an official proposal.
The latest ABC News/Washington Post poll found an overwhelming 83 percent of Americans support stricter border controls, including 64 percent who say they're "strongly" supportive. Fifty-five percent approve of a path to citizenship for illegal immigrants, with 41 percent opposed.
Still, immigrant advocates say public support for a comprehensive package of immigration reforms -- which would include stricter border controls and a pathway to citizenship -- has never before been so popular, promising to alter the assumed calculus for getting a bill through Congress.
"I think you're seeing a major change," said Brad Bailey, a Republican and executive director of the Texas Immigration Solution. "Once the specifics are spelled out and explained, people will start agreeing and understanding and comprehending there's a whole plan."
Many supporters of an overhaul plan also say the outcome of the 2012 presidential election, with unprecedented Hispanic voter turnout and support for Democratic candidates, amounts to a warning for Republicans about the perils of alienating the fastest-growing voter bloc.
"It's one thing to shoot yourself in the foot. Just don't reload the gun," said Republican Sen. Lindsey Graham of South Carolina earlier this month.
"I intend not to reload this gun when it comes to Hispanics. I intend to tear this wall down and pass an immigration reform bill that's an American solution to an American problem."
Obama has begun mobilizing his new independent political advocacy group -- Organizing for Action -- to shore up support for his immigration plan among members of Congress on both sides of the aisle, calling on the group's 12 million email subscribers to share their personal stories related to immigration.
"Just telling our stories would change people's minds," wrote Jose Magana, an undocumented immigrant and law school student, in the message. "This is exactly how we're going to persuade people across the country to get behind President Obama's plan for comprehensive immigration reform."
A "Share Your Story" website will collect the narratives, encouraging visitors to submit a video testimonial, still photo of themselves -- or both -- to further personalize the appeal. Organizing for Action later plans to showcase and distribute the stories through various media, an official said.
The strategy mimics an approach the White House and the Obama campaign used regularly over the past four years to try to leverage popular opinion of the president's initiatives to spur legislative action and bolster the impression of widespread appeal.
But opponents of the effort say economics, not emotions, will ultimately prevail in the immigration fight.
"If Republicans think getting on the amnesty bandwagon this late in the game is suddenly going to make Hispanics come flocking to the Republican Party, they really are going to go the way of the dinosaur," said Mehlman. "They're only going to hasten their own demise."
This story has been updated to correct description of the Senate immigration plan, which would include mandatory employment verification and a nationwide exit/entry system.