The position could also be precarious. Mr. Trump fired his first ambassador to the European Union, Gordon Sondland, after he testified to Congress during Mr. Trump’s first impeachment inquiry. The inquiry centered around charges of a “quid pro quo” in a push to have the Ukrainian government investigate the family of Joseph R. Biden Jr., then Mr. Trump’s likely 2020 election rival. Mr. Trump was accused of linking that demand to an official meeting between him and Ukraine’s newly elected president, Volodymyr Zelensky — as well as the continued flow of American military aid.

Mr. Sondland, who was involved in the effort to pressure Ukraine to announce investigations of Mr. Biden’s family, testified that “we followed the president’s orders” and that “everyone was in the loop.”

Mr. Trump had previously selected Mr. Puzder to serve as his labor secretary early in his first term, but Mr. Puzder’s nomination fell apart under bipartisan scrutiny of his business record and character. Democrats, unions and liberal groups criticized him for his remarks against workplace protections such as the minimum wage and overtime rules, and conservative publications seized on Mr. Puzder’s employment of an undocumented immigrant as his housekeeper. Republican senators ultimately turned against Mr. Puzder amid an accusation of spousal abuse that resurfaced during the confirmation process. Mr. Puzder has denied the allegation, and his ex-wife, Lisa Fierstein, retracted the claim of physical abuse.

If confirmed, Mr. Puzder would take over the ambassadorship from Mark Gitenstein, a veteran ambassador and longtime aide to Mr. Biden who first worked as a lawyer for him more than four decades ago. Mr. Gitenstein stepped down from his position on Jan. 17.