Kellyanne Conway, a senior adviser to Donald Trump, hasn’t easily forgotten how former Massachusetts Gov. Mitt Romney panned the president-elect during the campaign.

In the morning political-show circuit Sunday, she said it’s this barefaced disloyalty to Mr. Trump that should disqualify Mr. Romney from becoming US Secretary of State.

“People feel betrayed to think that Gov. Romney, who went out of his way to question the character and the intellect and the integrity of Donald Trump, now our president-elect, would be given the most significant Cabinet post of all, Secretary of State,” Ms. Conway told host Chuck Todd on NBC’s “Meet the Press.” “They feel a bit betrayed to think that you can get a Romney back in there after everything he did.”

Conway isn’t the only member of Trump’s inner circle to publicly slam the idea of Romney becoming Secretary of State, bringing the infighting reportedly occurring among the transition team into full view. And yet, the debate over Secretary of State appears to come down to one question: loyalty. In other words, can Trump tap a person who, while perhaps qualified, was one of his harshest critics throughout the campaign? Or should he instead go with a consistently loud and strong Trump surrogate?

Conway, former House Speaker Newt Gingrich, and former Arkansas Gov. Mike Huckabee have indicated loyalty should matter. Conway first started to publicly denounce Romney when she took to Twitter on Thanksgiving Day. In a series of tweets, she said she received a “deluge” of social media and private communications that warned against picking Romney.

“This is extraordinary, if you think about it,” wrote Peter Grier for The Christian Science Monitor the following day. “A woman who is clearly part of the Trump inner circle is using public rather than private means to communicate displeasure. It’s possible that she’s being so open because she thinks only drastic measures can derail Romney’s appointment. It’s also possible she’s doing it simply because she knows it’s the best way to get Trump’s attention.”

When host Martha Raddatz of ABC's “This Week” asked Conway on Sunday why she publicly aired her concerns, Conway said she had discussed the issue privately with Trump and would respect his decision. But she made clear she personally opposed Romney as Secretary of State.

“There was the ‘Never Trump’ movement and then there was Governor Mitt Romney,” said Ms. Conway. “He went out of his way to hurt Donald Trump."

During the campaign, Romney said Trump posed a fundamental threat to American democracy, as The Wall Street Journal reported.

“If we, Republicans, choose Donald Trump as our nominee, the prospects for a safe and prosperous future are greatly diminished,” he said in a March speech in Utah.

“Donald Trump is a phony, a fraud,” Mr. Romney said then. “His promises are as worthless as a degree from Trump University. He’s playing the members of the American public for suckers: He gets a free ride to the White House, and all we get is a lousy hat.”

But after Trump’s victory, he, Romney, and Vice President-elect Mike Pence appeared to put these remarks behind them. In a Nov. 19 meeting at Trump National Golf Club in Bedminster, N.J., Romney said he and Trump had a “far-reaching conversation.” Since then, Romney’s name has emerged alongside that of Rudolph Giuliani as the main contenders for the top cabinet post.

But the appearance of Romney's name on this short list wasn’t well received by Conway, Mr. Gingrich, or Mr. Huckabee. Gingrich, who has openly supported Mr. Giuliani, the former New York City mayor, said last week that Romney was the president-elect’s “most vicious and most explicit opponent all through the campaign on the Republican side.”

He added that Secretary of State isn’t the first choice of Romney, a 2008 and 2012 presidential candidate.

“Gov. Romney wanted to be president, not Secretary of State, and you have to ask the question: When he goes overseas, is he gonna be the Secretary of State for President Trump or is he gonna be Mitt Romney’s own secretary of state?” Gingrich asked.

The current and previous Secretaries of State contradict Gingrich's point. Both John Kerry, the acting secretary of State, and Hillary Clinton, ran, unsuccessfully, for president. In fact, Mrs. Clinton served under President Obama after he beat her in the 2008 Democratic primary.

Two of Trump’s other cabinet appointments this past week also didn’t show the president-elect enduring loyalty. Trump nominated South Carolina Gov. Nikki Haley to US ambassador to the United Nations, even though she had clashed with Trump over his comments about Mexicans and Muslims and endorsed Sens. Marco Rubio and Ted Cruz during the Republican primary, according to The Wall Street Journal. Trump also nominated Betsy DeVos as his Secretary of Education, even though she was a delegate for Ohio Gov. John Kasich at the Republican National Convention. Both cabinet positions require Senate confirmation.

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But Conway indicated this past week that loyalty is a foremost quality in secretary of State, tweeting that Henry Kissinger and George Schultz were secretaries of State who “flew around the world less, consulted POTUS close to home more. And were loyal. Good checklist.”

According to the Monitor’s Mr. Grier, however, Mr. Kissinger was not really any of those things.