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“I don’t think Speaker Boehner can pass piecemeal bills,” Mr. Schumer said.

Rep. Luis V. Gutierrez, Illinois Democrat and chairman of the House Hispanic Caucus Immigration Task Force, said House Republicans are casting aside the bipartisan work that the Senate has accomplished.

“What the House Republicans are doing is giving a Republican solution. A Republican solution isn’t what we saw was successful in the Senate,” he said, also appearing on CNN. “It’s a solution that’s going to lead us to nothing.”

Mr. McCain and other Senate Republicans who back the comprehensive package are nudging the House to action while acknowledging the difficult task Mr. Boehner faces.

“I believe Speaker Boehner has a tough job ahead. I admire his leadership, but we’ve got a lot of work to do,” said Mr. McCain, also appearing on Fox.