Gregory Korte

USA TODAY

WASHINGTON — President-elect Donald Trump is widening his search for a secretary of State after high-profile meetings with four top candidates failed to yield a decision last week.

"It is true that he's broadened the search, and the secretary of State is an incredibly important position for any president," Trump aide Kellyanne Conway told reporters at Trump Tower Sunday. "There is not a finite list of finalists only because he will interview with additional candidates early this week."

Last week, the transition team said the top diplomat's position had been "narrowed down" to four potential candidates: Senate Foreign Relations Committee Chairman Bob Corker, former New York City mayor Rudy Giuliani, retired Army general and former CIA director David Petraeus and 2012 GOP presidential nominee Mitt Romney.

Former United Nations ambassador John Bolton also met with Trump on Friday, and Vice President-elect Mike Pence included Bolton's name in a list of possible secretaries of State Sunday.

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Conway suggested that more candidates had come forward eager to join the Trump administration. "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, and they are coming forth now and expressing interest," she said.

The expanding list of candidates comes as one of the top candidates, Petraeus, appeared on ABC's This Week to answer questions about his interest in the job — and his conviction for mishandling classified information.

He pleaded guilty last year to a misdemeanor charge of giving journals containing state secrets to his biographer and mistress, Paula Broadwell. FBI Director James Comey has said that violation was far worse than what Trump's former rival, Hillary Clinton, was accused of in receiving classified information on a private email server. He remains on probation, meaning he needs approval from a probation officer — or a presidential pardon — in order to leave western North Carolina to serve in government.

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"Five years ago, I made a serious mistake. I acknowledged it. I apologized for it. I paid a very heavy price for it and I've learned from it," he said. "And, again, they'll have to factor that in and also obviously 38-and-a-half years of otherwise fairly — in some cases — unique service to our country in uniform and then at the CIA and then some four years or so in the business community."

Petraeus did not have an answer for how the Trump administration would fulfill a campaign promise of building a wall on the U.S.-Mexico border — and make Mexico pay for it. "I'm not sure what the scheme would be there for that," he said.

Pence defended Petraeus on This Week, calling him "an American hero,"

"But look, he made mistakes. And he paid the consequences of those mistakes." He said Trump would "factor the totality of Gen. Petraeus' career in making that decision."

"Whether it be Gen, Petraeus or Mitt Romney or Rudy Giuliani or Sen. Bob Corker or John Bolton or others who may be added to the list, what people are seeing is an extraordinary capacity of an executive to bring the broadest range of people around him," Pence said.