"We're not really talking about 11 million if you start adding all this," Carmona said, calling such reforms a "partial solution" that would lead to a more complex, intractable problem.

Enforcing immigration laws

Goodlatte said these are "legitimate questions" to raise about a pathway to citizenship. Still, he said the problem of illegal immigration will persist unless the laws on the books are fully enforced. Goodlatte said no administration, from Reagan's to Mr. Obama's, has done so sufficiently.

"None have really seriously, significantly enforced immigration law in a comprehensive way, it's been selective," he said. "It's important we address that aspect if we proceed in a larger sense."

Many immigrant advocates balk at the notion that there's lax enforcement -- while the president has allowed for some undocumented youths to stay in the country, U.S. Immigration and Customs Enforcement (ICE) has set deportation records under Mr. Obama's leadership. Meanwhile, the U.S. government is spending more on immigration enforcement than all other principal federal criminal law enforcement agencies combined, according to the Migration Policy Institute, and the number of undocumented immigrants has declined from its peak of 12 million in 2007.

Still, lawmakers from both parties seem ready to find common ground on this issue.

"In Texas, we know firsthand that this administration has put more boots on the ground along the border than at any time in our history, which has led to unprecedented success in removing dangerous individuals with criminal records," San Antonio Mayor Julian Castro, a Democrat, plans to tell the Judiciary Committee today, according to his prepared testimony. "But Democrats and Republicans can agree that the work to ensure America's safety and security is ongoing, and should be part of the legislative agenda going forward."

Enforcing the law means not only keeping the border secure but cracking down on employers for hiring undocumented workers. When it comes to deterring illegal immigration, "The biggest gap remains employer enforcement because people come here for jobs," Doris Meissner, senior fellow at the Migration Policy Institute, told CBSNews.com. The immigration reform blueprint unveiled in the Senate addresses this in part through a mandatory electronic employment verification system and revamping entry-exit system to more effectively curtail visa overstays.

Improving legal immigration

Since people come here for jobs, it's clear the legal immigration system must be revamped as well.

Vivek Wadhwa, director of research at the Pratt School of Engineering at Duke University, says in his prepared opening statement for today's hearing that the United States' visa policies are "chasing away" talent that's needed more than ever in the competitive global economy.

"Foreign students graduating from American colleges have difficulty in finding jobs because employers have difficulty in getting H1-B visas," his opening statement says. "Those graduates who are lucky enough to get a job and a visa and who decide to make the U.S. their permanent home find that it can take years--sometimes more than a decade--to get a green card. If they have ideas for building world-changing technologies and want to start a company, they are usually out of luck, because it is not usually possible for people on H1-B visas to work for the companies they might start."