A "safety valve" would allow employers to exceed the cap, the official said, if they could show need and pay premium wages, but any additional workers brought in would be subtracted from the next year's cap.
The workers could move from employer to employer and would be able to petition for permanent residency and ultimately seek U.S. citizenship. Neither is possible for temporary workers now.
"As to the 11 million (illegal immigrants), they'll have a pathway to citizenship, but it will be earned, it will be long, and it will be hard, and I think it is fair," Graham said.
The new program would fill needs employers say they have that are not currently met by U.S. immigration programs. Most industries don't have a good way to hire a steady supply of foreign workers because there's one temporary visa program for low-wage nonagricultural workers but it's capped at 66,000 visas per year and is only supposed to be used for seasonal or temporary jobs.
Separately, the new immigration bill also is expected to offer many more visas for high-tech workers, new visas for agriculture workers, and provisions allowing some agriculture workers already in the U.S. a speedier path to citizenship than that provided to other illegal immigrants, in an effort to create a stable agricultural workforce.
Schumer, Flake and Axelrod appeared on NBC's "Meet the Press." Graham was interviewed on CNN's "State of the Union."
Associated Press writer Erica Werner contributed to this report.
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