The wrangling over who should be America’s top diplomat comes as Mr. Trump’s team continues to face questions about his protocol-shattering phone conversation with the president of Taiwan last week, which angered China and rattled other Asian nations. Mr. Bolton advocates closer ties with Taiwan as a means of putting pressure on China, arguing that Beijing’s growing power in the region should be checked.

Mr. Trump’s advisers are battling, at times publicly, over whether he should choose from among his campaign loyalists or go outside that circle, a move that could alienate voters who were angry at the Republican establishment. Much of that wrangling has centered on Mr. Romney, who was among Mr. Trump’s fiercest critics during the campaign.

Mr. Trump has told aides that he believes that Mr. Romney would “look the part” as the face of American outreach to the world and would make a fine secretary of state.

But Ms. Conway, who has been openly critical of Mr. Romney, continued to attack him during a “Fox News Sunday” appearance, saying that the backlash to his candidacy among Mr. Trump’s core supporters had been “breathtaking.”

Asked if her criticism of him was appropriate, she said, “I would turn the question around and ask, was it appropriate for Governor Romney to stick his neck out so far in attacking Donald Trump, and never walking it back, never encouraging people to support the nominee once Mr. Trump had won the nomination squarely and fairly.”

Ms. Conway, answering questions from reporters about the expanded search as she entered Trump Tower in New York on Sunday, said the transition team had been happily surprised by the number of wealthy businesspeople who had come forward to express interest. Mr. Trump, who was lifted to victory in the election in large part by the support of white working-class voters, has faced criticism over his appointment of several ultra-wealthy people to top posts.

“There are a number of people that we may not have thought wanted to leave their very lucrative private industry positions to go and serve the government,” Ms. Conway said. “It’s exciting, frankly, to at least get their counsel.”