“It is true that he's broadened the search, and the secretary of state is an incredibly important position for any president to fill,” she said. “He's very fortunate to have interest among serious men and women.”

Asked about the search on ABC News's "This Week," Vice President-elect Mike Pence mentioned John R. Bolton, an ambassador to the United Nations under President George W. Bush, as a potential secretary of state candidate and said others could be added to the list.

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Among the other names that have surfaced for the position is Jon Huntsman, a former Republican Utah governor who also has served as ambassador to China. Huntsman sought the Republican presidential nomination in 2012, backing Romney after dropping out.

Speaking to reporters in Trump Tower, Conway declined to confirm consideration of Huntsman or to offer names of other possible contenders.

“Who knows how many finalists there will be,” she said. “It's a big decision, and nobody should rush through it.”

Besides Romney, the secretary of state finalists whom Trump aides had previously identified were former New York mayor Rudolph W. Giuliani, a Trump confidant during the campaign; retired decorated military officer David H. Petraeus; and Sen. Bob Corker (R-Tenn.), chairman of the Senate Foreign Relations Committee.

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Sen. Joe Manchin (D-W.V.) has also surfaced as a possibility in recent days, and on Sunday, The Wall Street Journal reported that Rex W. Tillerson, the president and chief executive of Exxon Mobil, could be in the mix.

Trump’s consideration of Romney has been most visible, including a dinner last week in New York that Romney afterward described as “a wonderful evening.”

The choice of Romney, the 2012 Republican nominee, would send a reassuring signal to the moderate wing of the party and to the foreign policy establishment. But Trump loyalists have bristled at the prospect of rewarding someone who was among Trump’s harshest Republican critics during the campaign.

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In arguing against Trump’s fitness for office, Romney called him a “phony,” a “fake” and a “con man.”

Evidence of the unease with Romney was evident at Trump’s first stop on a “thank you” tour last week in Cincinnati, where chants of “No Romney” broke out among the boisterous crowd.

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Giuliani has openly campaigned for the secretary of state post but faces questions about his international business ties that could make Senate confirmation challenging.

Petraeus, a former CIA director, also has interviewed with Trump, but he has a significant liability: Last year, he pleaded guilty to a misdemeanor charge of mishandling classified information, admitting that he gave sensitive materials to former lover and biographer Paula Broadwell.

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During an appearance Sunday on “This Week,” Petraeus acknowledged his mistake and said that “they'll have to factor that in and also obviously 38½ 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.”

Corker would probably be easily confirmed by his Senate colleagues but has said he does not expect to get the job.

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During an interview on CBS’s “Face the Nation,” Reince Priebus, Trump’s incoming chief of staff, acknowledged that the search “is taking just a little bit longer, and I think it’s just fine.”

“Everything doesn't have to happen all at once,” Priebus said. “He's taking his time, making a smart decision, and we will see where that goes.”

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Priebus sought to play down the significance of Romney’s past criticism of Trump.

“I think Governor Romney is very talented,” he said. “And I think what really we should look at is that we have got a president here in Donald Trump that wants to look at the best and brightest of America, regardless of background, regardless of past disputes that we may have had with each other, that that is the past. That's the rearview mirror.”

Conway, who has publicly voiced concerns about Romney, however, questioned during another television appearance on Sunday whether it was “appropriate for Governor Romney to stick his neck out so far in attacking Donald Trump” during the campaign.

“And never walking it back, never encouraging people to support the nominee once Mr. Trump had won the nomination squarely and fairly?” Conway said on “Fox News Sunday.”