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The Medicare administrator’s job carries major responsibilities under the health care law, such as setting up new insurance markets, expanding Medicaid to cover millions more low-income people, and revamping the way Medicare pays providers to reward quality instead of volume.

Mr. Obama bypassed the Senate last year to install Dr. Berwick as a recess appointment. Under the rules, that means that unless Dr. Berwick can be formally confirmed, his appointment would run out at the end of this year.