Ms. Noem, the governor of South Dakota, is a loyalist who has vocally supported Donald J. Trump’s pledges regarding undocumented immigrants.

Noem, a staunch ally of the president-elect, was previously reported to be among Trump’s vice presidential contenders. However, she appeared to damage her chances when she mentioned in her memoir, “No Going Back,” that she once shot and killed one of her dogs in a gravel pit.

The DHS secretary is key to Donald Trump’s domestic policy agenda. He has pledged mass deportations of undocumented immigrants and a crackdown at the U.S.-Mexico border.