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“I come to you as one of our country’s 11 million undocumented immigrants, many of us Americans at heart but without the right papers to show for it,” Mr. Vargas said. “Too often we’re treated as abstractions, faceless and nameless, subjects of debate rather than as individuals with families, hopes, fears and dreams.”

In winning re-election last year, Mr. Obama triumphed over Republican Mitt Romney, who laid out the toughest immigration policy of any major party nominee in history. Mr. Obama, meanwhile, campaigned on a promise of legalizing the 11 million illegal immigrants now in the U.S., and Hispanic voters turned out for him in overwhelming numbers.

Stung by those results, many Republicans have concluded that they must join Mr. Obama in his immigration efforts this year.

But others aren’t convinced that the country can succeed at legalizing 11 million people without inviting another wave of illegal immigration — which is what happened after the 1986 amnesty. That experience weighed heavily on senators.

Sen. Jeff Sessions, an Alabama Republican who opposes Mr. Obama’s call for legalization, said that “1986 is so fundamental. We need to see, the American people need to see a real commitment.”

Ms. Napolitano said that since 1986, the Border Patrol has grown sixfold, fencing has grown from a few miles of chain-link fence to more than 600 miles divided between pedestrian fencing and vehicle barriers, and annual deportations have grown from 25,000 to more than 400,000 in 2012.

If Congress wants to increase enforcement, she said, it should come from the Interior Department, which could prevent businesses from hiring illegal immigrants.

All sides agree that jobs are the magnet that draws most illegal immigrants to the country.

House Republicans have signaled that they would like to look at the immigration issue in pieces — though Majority Leader Eric Cantor, Virginia Republican, said last week that he supported at least legalizing young illegal immigrants who were brought to the U.S. by their parents.