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Mr. Gutierrez said in the end, he expects “the immense majority” of the estimated 11 million illegal immigrants in the country will qualify for legalization.

Anticipating the legislative fights to come, members of the “Gang of Eight” emphasized that their plan bolsters national security through the proposed E-Verify system and stronger border security and also ensures that American workers get the first crack at open jobs.

“The status quo threatens our security, damages our economy, disregards the rule of law and neglects our humanitarian responsibilities,” said Sen. John McCain, Arizona Republican. “A problem of that magnitude that affects so many of our interests will never be easy to address but never more necessary to address either.”

Sen. Marco Rubio, Florida Republican, also went on a public relations blitz, assuring conservative radio outlets that the proposal, among other things, does not lead to “amnesty.”

Mr. McCain also said the bill will go through the normal legislative process starting with committee hearings next month but that the eight senators are poised to act together and defeat any amendment that would kill the bill outright.

“We are also committed to vote against amendments or proposals or changes that would kill the bill,” Mr. McCain said. “We’re not saying it’s a perfect piece of legislation, and we think it can be improved on, but we also know opponents will be proposing amendments that, if passed, could collapse this very fragile coalition that we’ve been able to achieve.”

Mr. Gutierrez, too, predicted difficult sailing for the legislation, and he said that will be when Mr. Obama can best play a role.

“Look, there’s going to come some hard times and we’re going to need him to use that bully pulpit, I believe, in the future,” he said. “It’s going to get stymied, there’s going to be hiccups, you guys have been around long enough, we’re going to need him there, so he is critical and essential to this process.”