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“One of the big problems right now is that under the administration’s policy, people who are not approved for legal status are not a priority for removal — in other words, nobody is going after them to enforce the law. Whether it’s people who have been rejected for a green card or people who have been ordered removed by a judge, and absconded,” she said. “So even if there were to be a legalization program, there’s very little chance that people who don’t qualify under the rules Congress writes will have to leave. So it’s 100 percent amnesty anyway.”

On Sunday, Sen. Charles E. Schumer, New York Democrat, acknowledged the lack of trust and floated an idea designed to get around it: Pass the bill now but delay all of the implementation until 2017, when the next president is in office.

“I think the rap against [the president] that he won’t enforce the law is false. He’s deported more people than any other president. But you could actually have the law start in 2017 without doing much violence to it,” Mr. Schumer said on NBC’s “Meet the Press.”

He said the legalization program then would begin with anyone who arrived as of the end of last year — which would be more generous than the Senate bill, which cut off at the end of 2011. Mr. Schumer’s plan would mean almost every illegal immigrant in the U.S. now would be eligible for an eventual path to citizenship.

Still, that idea comes with several major questions, including the status of illegal immigrants in the intervening three years, whether their families would be welcome to join them during that time, and whether enforcement could be boosted quickly enough to make sure no more people gain access to the U.S. illegally.

Sen. Rob Portman, an Ohio Republican who voted against the Senate bill last year, said Mr. Schumer’s idea could attract some Republican interest as long as there was an effort to boost enforcement.

“The concern we have, as you know, is to get back to the 1986 law, last time we did this, where we did provide legalization but didn’t do the enforcement, 3 million people were legalized, another 6 million people came illegally,” he said.