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For his part, Mr. Rubio said he’s open to a piecemeal approach, but said in the end it will have to include enough parts to equal the whole of a comprehensive bill.

“This issue can be handled in separate bills, but ultimately it must be handled comprehensively,” he said.

Also Wednesday the Center for Immigration Studies, a think tank that wants to see strict controls on immigration, released a report arguing that the Senate bill would nearly double the number of guest-workers in the country, from nearly 700,000 in 2012 to more than 1.3 million by the time the legislation takes full effect.

Those on both the right and left have said they fear American workers would suffer if businesses were allowed to bring in temporary workers to compete for jobs.

Tom Howell Jr. contributed to this report.