Sen. Marco Rubio says a proposed immigration bill expected to be introduced this week won't offer amnesty to those who entered the U.S. illegally.

Senator Marco Rubio
Senator Marco Rubio (R) (NBC)
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The Florida Republican, who appeared on five news shows Sunday, says "there will be consequences for having violated the laws."

Rubio's proposal would require people to pass a "rigorous background check" and pay fines and application fees to receive a permit that would allow them to "work, travel and pay taxes." After 10 years they would be able to apply for legal immigration status and an eventual path to citizenship.

Under the proposal, the applicants would not be eligible for any federal benefits such as health care.

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