One person familiar with the current discussions said that Kelly Knight Craft, the ambassador to Canada, was being discussed as a possibility for the role. Others said that another name being floated was Richard Grenell, the ambassador to Germany who recently spent time with a United States delegation that included Ivanka Trump.

Although Mr. Trump had announced Ms. Nauert as his nominee for the ambassadorship, she had not been formally nominated. Her nomination was proceeding without a completed background check, according to two administration officials, and the State Department had not submitted the necessary paperwork to the Senate, raising questions about the delay among Republican Senate staff aides.

Some members of the Senate Foreign Relations Committee had expressed reservations in private about her candidacy, said Brett Bruen, a former diplomat and global engagement official in the Obama administration. The members had questions about whether she had the experience and expertise to deal with counterparts from other countries, especially ambassadors from Russia and China, the two main rivals of the United States on the United Nations Security Council.

“I had heard from multiple people on the Hill that her nomination was not going well,” Mr. Bruen said. “Heather’s background was not commensurate with the experience necessary for one of our most difficult diplomatic assignments.”

In a separate statement, Michael R. Pompeo, the secretary of state, described Ms. Nauert’s decision as personal. “I wish Heather nothing but the best in all of her future endeavors and know that she will continue to be a great representative of this nation in whatever role she finds herself,” he said.