New York, United States: Donald Trump on Tuesday stepped up his contentious search for a secretary of state that has split loyalists and sparked a backlash over candidates considered frontrunners to be America's next top diplomat.

The president-elect's nominee for the job will be America's public face to the rest of the world, the person who will succeed John Kerry, head up a department of 70,000 staff and lead the largest diplomatic operation in the world.

The prospective candidates touted most frequently have been erstwhile Trump critic and former Massachusetts governor Mitt Romney, celebrated general yet scandal-clad former CIA director David Petraeus, Senate Foreign Relations Committee chairman Bob Corker and outspoken former New York mayor Rudy Giuliani.

Trump held talks Tuesday with Senator Corker, considered a safe pick without the baggage of Petraeus and free from Romney's history of disloyalty to the billionaire.

Corker, 64, said after the meeting that he thought Trump had narrowed the choice "to a very small group of people" and it was important that the president-elect selects somebody on the same wave length.

"This is a decision that he needs to make," the Tennessee senator told reporters. "He needs to choose someone that he's very comfortable with and he knows that there's going to be no daylight between him and them."

He said that Trump did not discuss a timeline for announcing his pick. "I think he'll make the decision when he's comfortable," said Corker.

Romney, 69, a favorite among establishment Republicans, is scheduled to meet Trump later on Tuesday for a second face-to-face meeting with the president-elect in 10 days.

- Petraeus 'premature' -

But some among Trump's inner circle are horrified at the prospect of rewarding with such a plum job an outspoken critic who castigated Trump during the campaign as a "fraud" and a "conman."

His distrust of Russia -- at odds with a president-elect who has spoken admiringly of Vladimir Putin -- would also reassure establishment Americans.

Best known as the Republican nominee who lost the 2012 election to Barack Obama, Romney managed the 2002 Winter Olympics in Salt Lake City and was a successful businessman before going into politics.

Petraeus, who emerged latterly as a potential candidate and who met the president-elect at Trump Tower in New York on Monday, has by far the most foreign policy experience of the lot.

"Very impressed!" tweeted Trump just minutes after the meeting. The 64-year-old scholar-warrior commanded troops in Iraq and Afghanistan, was credited with masterminding the Iraq surge and became head of the CIA in 2011.

But his glittering public career came crashing to a halt in 2012 when he resigned from the CIA after showing classified material to his mistress and biographer Paula Broadwell. 

In 2015, he pleaded guilty to a misdemeanor charge of mishandling classified materials, and was put on two years' probation and fined $100,000.

Trump spokesman Jason Miller told Fox News Radio that it "might be a little bit premature" to assume Petraeus is being considered.

His appointment would arouse accusations of hypocrisy after Trump savaged Hillary Clinton for mishandling classified emails, but experts do not believe the scandal is insurmountable given the respect he commands on Capitol Hill.

- Twitter rants -

Republican Senator Lindsey Graham told reporters on Tuesday that he was "a big fan" of Petraeus. Republican Senator John McCain, another Trump critic, said he would "love" to see him return to public service.

Democratic Senator Dianne Feinstein, also told reporters she had "great respect" for the general.

Giuliani, who campaigned tirelessly for Trump, has been another contender, but scrutiny over business dealings has raised questions that could disqualify him and his penchant for rhetoric may sit uneasily with becoming chief diplomat.

On Tuesday, Trump expanded his cabinet further by nominating a fierce Obamacare critic as health secretary -- Congressman and former surgeon Tom Price -- indicating that he plans to tear up the divisive healthcare law.

US media reported that he had also selected Elaine Chao, the Taiwan-born former labor secretary as transportation secretary.

In between back-to-back job interviews, Trump has fanned alarm and inflamed critics by indulging in his customary tweet storms.

He sparked uproar by saying nobody should be allowed to burn the US flag -- allowed under the US constitution that safeguards freedom of expression.

He also embarked on a rant complaining about CNN and widespread voter fraud, which experts and elected officials deny happened.