WASHINGTON — Rival factions of Republicans are locked in an increasingly caustic and public battle to influence President-elect Donald J. Trump’s choice for secretary of state, leaving a prominent hole in an otherwise quickly formed national security team that is unlikely to be filled until next week at the earliest.

The debate inside Mr. Trump’s wide circle of formal and informal advisers — pitting supporters of one leading contender, Mitt Romney, against those of another, Rudolph W. Giuliani — has led to the kind of dramatic airing of differences that characterized Mr. Trump’s unconventional and often squabbling campaign team.

And it traces the outlines of the enduring split in the Republican Party between establishment figures who scoffed at Mr. Trump’s chances of victory and the grass-roots insurgents who backed him as a disrupter of the Washington power structure.

The most publicly vocal faction has been the group opposed to Mr. Romney, which has questioned whether he would be loyal after his searing criticism of Mr. Trump during the campaign. But Mr. Trump himself has told aides that he believes Mr. Romney “looks the part” and would make a fine secretary of state, a senior Trump official said on Thursday. Mr. Trump, who is always difficult to read and is capable of changing his mind at any minute, has also praised Mr. Giuliani in recent conversations with acquaintances.

Even Thanksgiving did not provide a reprieve from the extraordinary public efforts to cast doubt on Mr. Romney. Mr. Trump’s campaign manager, Kellyanne Conway, said on Twitter that she had received “a deluge” of concern from people warning against picking Mr. Romney, the 2012 Republican presidential nominee and former Massachusetts governor.

Those raising concerns about Mr. Giuliani, the former New York City mayor and an early and loyal supporter of Mr. Trump, have said they fear that his tangle of foreign business ties could lead to a damaging confirmation battle. They are also worried that Mr. Giuliani lacks the vigor for the globe-trotting job.

Both Mr. Romney and Mr. Giuliani have made their interest in the role known to Mr. Trump. But while Mr. Giuliani has been very public about his intentions — angering Mr. Trump at times with his statements — Mr. Romney has been more reserved.

Gen. John Kelly, the former head of the United States Southern Command, has emerged as an alternative to Mr. Romney and Mr. Giuliani.© Mandel Ngan/Agence France-Presse — Getty Images Gen. John Kelly, the former head of the United States Southern Command, has emerged as an alternative to Mr. Romney and Mr. Giuliani.

The split over the two men has opened the door for another candidate altogether. One potential pick Mr. Trump and his team have entertained is Gen. John Kelly of the Marines, a former head of the United States Southern Command. Others are David H. Petraeus, the retired general and former C.I.A. director, and Senator Bob Corker of Tennessee, according to two people involved in the process.

Asked about Mr. Trump’s deliberations, a spokesman, Jason Miller, said in an email Thursday, “The president-elect is meeting with a number of well-qualified potential selections for this important position who share his America First foreign policy — some of whom have been made public and others who have not — and the president-elect will make public his decision when he has finalized it.”

Mr. Romney would represent a departure from the hard-liners Mr. Trump has already picked for his national security team. But aides like Stephen K. Bannon, Mr. Trump’s chief strategist, have expressed doubts about how loyal Mr. Romney would be given his denunciation of Mr. Trump as a “phony” and a “fraud.” Mr. Bannon and others have told colleagues they fear that a State Department under Mr. Romney could turn into something of a rogue agency.

Asked to explain her Twitter post about Mr. Romney, Ms. Conway said that while she trusted Mr. Trump’s judgment, she found it notable that the most outrage directed at Mr. Trump from the party’s grass-roots “is not against something he said, but something he may do.” In another post, she said that being “loyal” was an important characteristic for a secretary of state.

Others hoping to catch Mr. Trump’s ear have taken their message to a place they know he is likely to absorb it: cable news. Joe Scarborough, the MSNBC host, who has spoken with Mr. Trump about his concerns that Mr. Giuliani would not be confirmed by the Senate, has taken to making those arguments on a daily basis on his morning show, which he knows Mr. Trump watches.

Others, like Newt Gingrich, the former Republican House speaker, and Mike Huckabee, the former governor of Arkansas, have gone on television to try to dissuade Mr. Trump from picking Mr. Romney. Mr. Huckabee, who said during the 2008 presidential campaign that Mr. Romney reminded voters of “the guy who laid them off,” told Fox News on Wednesday that picking Mr. Romney would be “a real insult” to Mr. Trump’s supporters. Mr. Giuliani is a favorite of the Republican voters who turned out in large numbers to lift Mr. Trump to victory.

Sean Hannity, a Fox News host whose opinion Mr. Trump often privately solicits, has also been deeply critical of Mr. Romney on his show.

Shortly after the election, Mr. Giuliani told associates that he believed the job was his. He had communicated to Mr. Trump’s top advisers that it was the only post he was interested in, according to the people briefed on the discussions.

But he began to run afoul of Mr. Trump when he told a Wall Street Journal forum that he would probably be a better candidate than John Bolton, who served as one of George W. Bush’s ambassadors to the United Nations.

And when reports surfaced about Mr. Giuliani’s foreign business entanglements and highly compensated speechmaking, Mr. Trump grew even warier. His firm, Giuliani Partners, has had contracts with the government of Qatar, and Mr. Giuliani has given paid speeches to a shadowy Iranian opposition group that until 2012 was on the State Department’s list of foreign terrorist organizations.

As a backup plan, some of Mr. Trump’s aides encouraged him to meet with Mr. Romney. Though some in Mr. Trump’s inner circle, like Reince Priebus, his choice for chief of staff, thought that such a meeting would anger the president-elect’s supporters, Mr. Trump went ahead, and he started sounding Mr. Giuliani out about a different post, director of national intelligence.

But what many people believed would be a perfunctory meeting with Mr. Romney last weekend at Mr. Trump’s golf club in Bedminster, N.J., turned into something more substantial.

Mr. Trump liked Mr. Romney quite a bit, and was intrigued by the possibility of such a camera-ready option to represent the country around the globe, advisers to Mr. Trump said. The following day, Mr. Giuliani met with Mr. Trump and urged him to make a decision in one direction or the other.

Mr. Romney, who was mocked in 2012 when he described Russia as the greatest geopolitical foe of the United States, has seen his stock in the Republican Party rise since his loss to President Obama, although he is still viewed skeptically by the party’s grass-roots. His allies believe that his position on Russia has been vindicated, but it is starkly at odds with Mr. Trump’s stated desire for a better relationship with the Russian president, Vladimir V. Putin.

Privately, Mr. Giuliani has expressed his frustration at going from front-runner for secretary of state to a contender who has to convince Mr. Trump of his strengths. He is particularly irritated over the focus on his business ties.

The option of a third person like General Kelly has gained currency in recent days inside the transition team. A respected leader, General Kelly served as the senior military assistant to former Defense Secretary Leon E. Panetta. He led the Southern Command, responsible for all United States military activities in South and Central America, for four years under Mr. Obama. And his appointment would fit Mr. Trump’s inclination toward putting people with combat experience in senior foreign policy roles.

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