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Sen. Rand Paul, Kentucky Republican, has put forward a proposal that calls for an independent auditor to evaluate whether the borders are secure and then have Congress vote on whether to accept that report.

Under his plan, illegal immigrants could earn “probationary status,” but would not be given a special path to citizenship.

The movement comes after Mr. Obama won over Hispanic voters in a landslide in the 2012 election, promoting Republicans to rethink their approach to immigration.

“Four years ago, when I started here and I went around the Hill saying, ‘Let’s work together on immigration reform,’” Ms. Napolitano said Tuesday. “I didn’t really get a positive response.”

“I mean, there were two wars going on, we were close to a depression, health care was winding its way through Congress and it was like, ‘We can’t take on another big issue.’ I think now is the time and I think the election had consequences,” she concluded.