NEW YORK — Donald Trump’s hunt for a secretary of state is veering into dramatic terrain, with the president-elect summoning Mitt Romney back for a second look as a top aide leads a public pressure campaign against the pick.

Trump has a follow-up meeting Tuesday with the 2012 GOP presidential nominee, who has become a symbol of the internal divisions agitating the transition team. He also plans to sit with Tennessee Sen. Bob Corker, the chairman of the Senate Foreign Relations Committee. On Monday, Trump spent an hour with retired Gen. David Petraeus, a new contender.

Aides were divided over Trump’s choices, particularly the prospect that Trump could tap Romney for the top-tier diplomatic post. In an unusual public airing of internal machinations, Trump senior adviser Kellyanne Conway on Sunday warned that the president-elect’s supporters would feel “betrayed” if he tapped Romney as secretary of state.

Exiting Trump Tower on Monday night, Vice President-elect Mike Pence simply teased “a number of very important announcements tomorrow.”

On Tuesday morning, the Trump transition team announced the selection of Georgia Rep. Tom Price to lead the Department of Health and Human Services, a choice that immediately drew Democrats’ fire. Price will be Trump’s point person on replacing President Barack Obama’s health care law.

Incoming Senate Democratic leader Chuck Schumer said Price “has proven to be far out of the mainstream of what Americans want” for programs that help seniors, women, families and those with disabilities. His nomination, Schumer said, is “akin to asking the fox to guard the hen house.”

Trump’s team also announced Tuesday that Seema Verma has been chosen to be administrator of the Centers for Medicare and Medicaid Services.

Jason Miller, a transition team spokesman, said at least one other Cabinet post would be announced in the afternoon. He did not elaborate.

While other staffing decisions were being made, the search for secretary of state was still underway. Petraeus said he spent about an hour with the president-elect and praised him for showing a “great grasp of a variety of the challenges that are out there.”

The former CIA chief pleaded guilty last year to a misdemeanor charge of mishandling classified information relating to documents he had provided to his biographer, with whom he was having an affair.

Pence, who is heading the transition effort, is said to be among those backing Romney for secretary of state. Romney was fiercely critical of Trump throughout the campaign but is interested in the Cabinet position, and they discussed it during a lengthy meeting earlier this month.

Other top Trump allies, notably Conway, have launched a highly unusual public campaign against a Romney nomination. Conway’s comments stirred speculation that she is seeking either to force Trump’s hand or give him cover for ultimately passing over Romney.

Three people close to the transition team said Trump had been aware that Conway planned to voice her opinion, both on Twitter and in television interviews. They disputed reports that Trump was furious at her and suggested his decision to consider additional candidates instead highlighted her influence.

Trump is said to have offered Conway a choice of White House jobs — either press secretary or communications director. But people with knowledge of Conway’s plans say she is more interested in serving as an outside political adviser, akin to the role President Barack Obama’s campaign manager David Plouffe played following the 2008 election.

Trump was considering former New York City Mayor Rudy Giuliani to head the Homeland Security Department, according to those close to the transition process. Giuliani was initially the front-runner for secretary of state and is still in the mix. But questions about his overseas business dealings, as well as his public campaigning for the job, have given Trump pause.

Those close to the transition insisted on anonymity in commenting because they were not authorized to publicly discuss the private process.

In addition to the public staffing spat, the campaign was forced Monday to defend Trump’s baseless assertions of illegal voting, made in angry response to a recount effort.

That effort, led by Green Party candidate Jill Stein and joined by Hillary Clinton’s campaign, marched on in three states, based partly on the Stein campaign’s unsubstantiated assertion that cyber-hacking could have interfered with electronic voting machines. Wisconsin officials approved plans to begin a recount as early as Thursday. Stein also asked for a recount in Pennsylvania and was expected to do the same in Michigan, where officials certified Trump’s victory Monday.

Trump has denounced the recounts and now claims without evidence that he, not Clinton, would have won the popular vote if it hadn’t been for “millions of people who voted illegally.” On Twitter, he singled out Virginia, California and New Hampshire.

There has been no indication of widespread election tampering or voter fraud in those states or any others. Trump aides struggled Monday to back up their boss’ claim.

Pace reported from Washington. AP writer Jonathan Lemire contributed to this report.