"There are a group of us that have met extensively to try to solve these issues," says Diaz-Balart. "I think the timing is right to really push it now in a renewed effort. We have a window of opportunity."
Diaz-Balart says he'd prefer to see Congress deal with immigration comprehensively, not piece by piece.
"I don't think we can do this piecemeal," he says. "This is a very complex and emotional issue on all sides. I think there is a realization that we have to confront the big problem that there are 12 million people living here illegally, and we cannot deport 12 million people."
No details have been finalized, but Diaz-Balart says the plan has to be bipartisan for the good of the country.
"Obamacare passed without bipartisan support. Forget about the merits. It creates this really bad feeling of distrust. On an issue this emotional, we have to lower the passion and get to the real work of creating reasonable, real, concrete solutions," he says.
Hutchison agrees that no matter what the details are, Congress will have to forge a deal with both sides of the aisle.
"If it is successful, it will be bipartisan. If it is partisan, it will not be successful," she says.
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Lauren Fox is a political reporter for U.S. News & World Report. She can be reached at firstname.lastname@example.org or you can follow her on Twitter @foxreports.