Andrew F. Puzder, who as President Trump’s pick to head the Labor Department has come under fire for criticizing worker protections, acknowledged on Monday that he had employed an undocumented immigrant to clean his house.

The revelation potentially could derail a nomination that has elicited controversy because of Mr. Puzder’s views on overtime pay, sick leave and automation. The confirmation of Mr. Puzder, a fast-food chain executive, has stalled as he tries to divest holdings.

“My wife and I employed a housekeeper for a few years, during which I was unaware that she was not legally permitted to work in the U.S.,” Mr. Puzder said in a statement. “When I learned of her status, we immediately ended her employment and offered her assistance in getting legal status.”

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