How important is loyalty?

That appears to be the question under public debate by members of Donald Trump’s transition team, as the president-elect ponders extending an offer to 2012 Republican presidential nominee Mitt Romney to serve as his secretary of state.

Romney served as Trump’s chief critic within the Republican party during the primary process, labeling the eventual nominee a “fraud” and a “phony” and campaigning for his opponents in critical battleground states.

On Saturday, the two men met at Trump’s golf course in New Jersey, where they discussed Romney coming on board in the Trump administration as secretary of state.

But that possibility wasn’t welcomed by some of Trump’s inner circle. On Thursday morning, campaign manager Kellyanne Conway tweeted that she had received a “deluge of social media and private comms [sic]” warning about Romney.

A subsequent tweet listed qualities she would like to see in a secretary of state, and notably included that the selection be “loyal.”

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The former Massachusetts governor’s primary competition for the job appears to be former New York City mayor Rudy Giuliani, a Trump loyalist and one of his closest advisors.

On Fox News in recent days, fellow long-time Trump backers Mike Huckabee and Newt Gingrich have each expressed strong reservations about adding Romney to the team. Huckabee has questioned why Romney would want the job, given the two men’s differences on several key foreign policy issues.

Gingrich, whose own presidential ambitions were dashed thanks in large part to a multi-million-dollar bombardment by a pro-Romney super PAC in 2012, has been more vocal in his preference for anyone but Romney.

“I would suggest there are a lot of other people who are more qualified than Romney in foreign policy and who also have not been as actively hostile as he’s been,” Gingrich told Fox.

On social media, former Romney staffers and supporters have publicly praised the consideration of Romney as a potentially moderating force in a Trump administration.

Dr. Ben Carson, a Trump rival-turned-supporter, when asked about what a Romney selection would show about Trump, offered this response on Fox News: “It shows that president-elect Trump is perhaps not as thin-skinned as they think he is.”