President-elect Donald Trump and his transition team on Friday are expected to turn their attention back to building out his Cabinet as controversy and speculation over his secretary of state pick continues.

Republicans have been locked in a public battle over whether onetime Trump critic Gov. Mitt Romney or former New York Mayor Rudy Giuliani should be nominated for the position.

The secretary of state, which must be confirmed by the Senate, is the president’s chief foreign affairs adviser and is tasked with carrying out the president’s foreign policies through the State Department.

Trump spokeswoman Kellyanne Conway went on Twitter Thursday and seemed to state an argument against the placement of Romney in Trump’s Cabinet. Romney vehemently opposed Trump's nomination early in the campaign, assailing the billionaire as a "phony."

In one tweet, Conway noted that she has been "receiving a deluge of social media & private concerns re: Romney Some Trump loyalists warn against Romney as sec of state."

In another, she makes references to former secretaries of state Henry Kissinger and George Schultz, both Cabinet secretaries in Republican administrations, as men who "flew around the world less, counseled POTUS (president of the United States) close to home more. And were loyal. Good checklist."

Former House Speaker Newt Gingrich and former Arkansas Gov. Mike Huckabee have both been skeptical about Romney.

“I think that if you want someone who is going to go out and be a very tough negotiator for America and represent American interest in the way that Trump campaigned, I think that probably Rudy is a better pick and has the right temperament,” Gingrich told reporters at Trump Tower earlier this week.

Following Trump’s meeting with Romney, Huckabee tweeted – but then later deleted - “I think there are huge advantages to Rudy Giuliani. Frankly, I think “Nice to have meeting -- but no admin spot; Mitt said Trump was con man; phony. That’s insult to Trump voters.”

But there are some Republicans who fear Giuliani will have a tough time getting through a Senate confirmation because of his foreign business ties.

Both Romney and Giuliani have made their interest in the job known to Trump. Giuliani has been public about his aspiration while Romney has been more reserved.

Before the Thanksgiving holiday, Trump announced he had tapped South Carolina Gov. Nikki Haley to serve as U.S. ambassador to the United Nations and charter school advocate Betsy DeVos to lead the Education Department.

The Associated Press contributed to this report.