Mr. Puzder added that he and his wife had paid back taxes in full to the I.R.S. and the state of California.

Confirmation hearings for Mr. Puzder have been delayed as he works to remove himself from his role as the chief executive of CKE Restaurants, whose holdings include the Carl’s Jr. and Hardee’s chains.

Democrats have suggested that his nomination could be on shaky ground, in part, because he has yet to file standard disclosures and questionnaires to the Senate committee that will consider his confirmation.

Progressive groups have echoed that sentiment. “It’s long past time for Mr. Puzder to deliver on his commitment to file paperwork with the Senate and let the American people know what issues are causing so many headaches for the Trump administration,” Karl Frisch, executive director of Allied Progress, said on Monday.

Mr. Puzder remains “all in” for the confirmation process, a spokesman said on Monday.

He is not the first cabinet nominee who has hired undocumented workers.