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Meanwhile, the House Judiciary Committee will work Thursday on two smaller bills — one that would impose a nationwide electronic worker verification system and another that would create a guest-worker program for agriculture workers.

Mr. McCaul said he believes his border security bill can pass the House and will eventually become part of any final conference negotiation with the Senate, if senators can pass a bill over there.

“I feel very confident we can pass this bill out of the House, and that puts us in play in a conference situation,” he said. “I think we’re all in agreement that more has to be done, that’s why we have this bill.”