On Wednesday, Mr. Trump rolled out their appointments with unstinting praise. Ms. Haley, he said in a statement, was “a proven deal maker, and we look to be making plenty of deals.” Ms. DeVos, he said, was a “brilliant and passionate education advocate.”
Credit Travis Dove for The New York Times
For Mr. Trump, who was spending a quiet day before Thanksgiving at Mar-a-Lago, his estate in Palm Beach, Fla., the appointments ended a breathless two-week sprint since his stunning victory. In a videotaped holiday greeting to the American people on Wednesday, he acknowledged the wounds left by a “long and bruising political campaign.”
His aides said he would resume meeting with potential cabinet members on Friday, when they said he would announce Mr. Carson, the retired neurosurgeon with whom Mr. Trump bitterly clashed and lavishly praised during the campaign.
Soon afterward, Mr. Trump is expected to name Gen. James N. Mattis as defense secretary. But the search for a secretary of state has become less clear, people involved in the transition said. Aides are divided between Rudolph W. Giuliani, who staunchly backed Mr. Trump’s candidacy but whose business dealings have drawn scrutiny, and Mitt Romney, viewed by many as a safe pick but who harshly criticized Mr. Trump during the campaign.
Mr. Giuliani, who has lobbied publicly for the position, confided to associates that he believed the job was his and told Mr. Trump’s top advisers that it was the only post he was interested in, according to the people briefed on the discussions.
But Mr. Trump grew concerned both with the attention Mr. Giuliani has been drawing to himself and the reports about potential business conflicts. Some in the president-elect’s circle voiced concerns about whether Mr. Giuliani would be the subject of a messy confirmation battle.
Their attention turned to an unlikely alternative: Mr. Romney. Advisers who favored Mr. Giuliani believed Mr. Trump could send a unifying message by holding a perfunctory meeting with a vocal critic like Mr. Romney. But Mr. Trump liked him and was intrigued by the prospect of a camera-ready emissary of the United States around the world.
Mr. Giuliani met with Mr. Trump a day later, urging him to make a decision. Mr. Romney has not signaled to Mr. Trump directly that he would accept the job if offered, a person close to the transition said.
Mr. Trump has also been warned by several advisers against choosing Mr. Romney because he might pursue his own agenda.
The tension has left some on the team looking for a third choice, like Gen. John F. Kelly of the Marines, the former head of the United States Southern Command, which oversees the military prison at Guantánamo Bay, Cuba; David H. Petraeus, the retired Army general who was director of the C.I.A.; or, a particular long shot, former Senator Jim Webb of Virginia, who could be the lone Democrat in the cabinet after having briefly run for the party’s presidential nomination last year.
Choosing Mr. Petraeus, who resigned in 2012 in a scandal over his leaking of classified information, would be somewhat easier after Mr. Trump said he would not pursue a case against Hillary Clinton over her handling of classified emails on a private email server, one person close to the transition said.
The debate over secretary of state is a microcosm of the balance Mr. Trump is trying to strike in all of his appointments — between loyalists who will appeal to his base and more mainstream Republicans, many of whom stridently opposed his candidacy. In Ms. DeVos, Ms. Haley and Mr. Carson, he is trying to appeal to several of these constituencies, from Midwestern ultraconservatives to the diverse society that Republicans have tried to reach, with mixed results.
Mr. Trump’s selections could blunt criticism that his early picks came from a homogeneous bloc of older, white men. If confirmed, Ms. Haley would step down as governor and be replaced by the state’s lieutenant governor, Henry McMaster, who was an early supporter of Mr. Trump.
Credit Hilary Swift for The New York Times
Mr. Carson is a familiar face to Americans after a Republican primary campaign in which he briefly rose to the top of the polls.
Neither he nor Ms. Haley is particularly experienced for the posts they have been offered. Mr. Carson had even seemed to take himself out of the running for a cabinet position last week, with his friends putting out word that he had concluded he was not qualified to run a vast federal bureaucracy. Some pointed to Ms. Haley’s experience as a legislator and trade ambassador for South Carolina as credentials for the United Nations post.
John D. Negroponte, a former ambassador to the United Nations under President George W. Bush who opposed Mr. Trump’s candidacy, said Ms. Haley was an “intriguing and very good choice.”
“She’s not an ideologue,” he said. “She’ll quickly grasp that there are aspects of the United Nations’ work that can benefit the United States.”
Ms. DeVos, a former finance chairwoman of the Republican National Committee, won the support of Senator Lamar Alexander, the Tennessee Republican who is chairman of the Senate Education Committee. He called her an “excellent choice.”
She favors charter schools, which are publicly funded but typically run independently of local school boards and teachers’ unions, and school vouchers, which give students tax dollars to apply toward private-school tuition.
Lily Eskelsen Garcia, president of the National Education Association, the country’s largest teachers’ union, criticized the nomination, saying that Ms. DeVos’s efforts over the years had “done more to undermine public education than support students.”
“She has lobbied for failed schemes, like vouchers — which take away funding and local control from our public schools — to fund private schools at taxpayers’ expense,” she said. “These schemes do nothing to help our most vulnerable students while they ignore or exacerbate glaring opportunity gaps.”