The field of candidates for secretary of State in Donald Trump Donald John TrumpOmar fires back at Trump over rally remarks: 'This is my country' Pelosi: Trump hurrying to fill SCOTUS seat so he can repeal ObamaCare Trump mocks Biden appearance, mask use ahead of first debate MORE's administration is "wide open," according "Morning Joe."

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Mitt Romney is at the top of the president-elect's list, sources familiar with Trump's thinking told the show.

Former New York City Mayor Rudy Giuliani is fading as a candidate, they added.

Jon Huntsman, a former Republican governor of Utah and U.S. ambassador to China, is reportedly not a serious contender to lead the State Department, while retired Gen. David Petraeus is also no longer a serious candidate.

Trump will meet Tuesday with ExxonMobil CEO Rex W. Tillerson and is "intrigued" with Tillerson's "vast knowledge of the world," according to "Morning Joe."

Trump campaign manager Kellyanne Conway said Sunday that Trump had "broadened the search" for the nation's top diplomat.

Trump has met with multiple candidates for the post throughout the transition.

The president-elect has faced backlash over his consideration of Romney for secretary of State.

Conway and former Speaker Newt Gingrich have both been critical of Romney, who spoke out strongly against Trump during his presidential campaign.