Sen. Bill Hagerty, R-Tenn., who served as ambassador to Japan during the first Trump administration, is another name that has been mentioned, though Republicans have only a three-seat majority in the upper chamber.
Richard Grenell, Trump's envoy for special missions, told Newsmax Thursday he is a "hard no" on being the U.S. ambassador to the United Nations.
"That's not something I want to do," Grenell said on "Greg Kelly Reports." "I've got plenty to do. President Trump has me doing different things. It's not someplace that I want to go to."
Morgan Ortatus, a former State Department spokesperson who works on Middle East issues, is said to also not be interested, Politico reported.