The split has put traditional Republican operatives such as Reince Priebus -- named Trump's chief of staff Sunday -- against more non-traditional influences such as Steve Bannon -- the alt-right leader of Breitbart News -- who will be Trump's chief strategist. A particular challenge is lack of clarity about the division of power among Priebus, Bannon and Trump's son-in-law Jared Kushner, who also has a key role in transition decisions. Another source tied to the transition described the resulting confusion as "buffoonery."
On Tuesday morning, former congressman Mike Rogers announced in a statement that he parted ways with the Trump transition team. Rogers' participation was seen as a heartening sign to many of the establishment side of national security advisers.
Vice President-elect Mike Pence, the new leader of the transition operation, will be at Trump Tower in New York Tuesday to meet with Trump and discuss possible nominees.
The divisions are being played out as Trump considers key appointments in the national security and diplomatic sectors, including secretary of state, with mainstream conservatives supporting John Bolton against Rudy Giuliani, who is seen as a loyalist to Trump.
On other key national security appointments, there is more agreement. Sen. Jeff Sessions is now the leading contender for attorney general, and is in the mix for secretary of defense as well, say multiple sources with knowledge of the transition.
Retired Lt. General Ron Burgess, former director of the DIA, is a leading contender for director of national intelligence. Retired General Michael Flynn is leading candidate for national security adviser.
No matter who gets what job, one description seems to fit the makeup of the short list: unconventional. And it highlights the dilemma faced by Trump, who is now torn between a campaign promise to shake up Washington and a need to build a national security team with policy experience.
The presence of so many political backers could signal that Trump values loyalty over experience and is keen to have people on board who share his worldview and are willing to depart from conventional wisdom.
This is particularly true when it comes to the secretary of state post. The position of America's chief diplomat is considered a particularly high-profile one, charged with maintaining relationships around the globe.
But some of the names being rumored for the job seem to indicate that Trump will favor an iconoclastic approach to the role:
Rudy Giuliani, possible secretary of state
Giuliani, a former federal prosecutor and mayor of New York, has little foreign policy experience to speak of but has been a vocal advocate of Trump's since early on in the campaign.
During his own run for the presidency in 2008, Giuliani espoused a fairly conventional Republican foreign policy view, calling on continued US engagement abroad and robust efforts to fight terrorism and stabilize Iraq and Afghanistan.
He also signaled support for democracy promotion, something at odds with Trump's pronouncements where he has put value on stability and counter-terrorism as opposed to democracy and human rights.
A source familiar with the transition process told CNN that Giuliani has expressed his preference for the secretary of state position as opposed to the post of attorney general.
John Bolton, possible secretary of state
Bolton, a diplomat in the George W. Bush administration, backed Trump soon after he secured the nomination and would be one of the more conventional choices.
But despite spending years at the State Department, Bolton has at times ruffled feathers in Washington. He was appointed to be ambassador to the UN, a Cabinet level position, in 2005. But his nomination was met with fierce opposition in Congress and Bush was forced to appoint him during a congressional recess.
A source familiar with the transition said the Trump team acknowledges that Bolton's confirmation might be difficult but they think it would be less tough than it was 10 years ago.
He also has elements of the conservative mainstream in his corner; he was endorsed by the influential National Review magazine in an open letter to President-elect Trump over the weekend.
Bolton has consistently advocated for unilateral and interventionist approaches to foreign policy challenges.
Like Trump he has opposed the Iran nuclear deal going so far as to call for airstrikes against Iran, penning a 2015 op-ed in the New York Times titled "To Stop Iran's Bomb, Bomb Iran."
He has also been highly critical of international institutions and organizations like the UN, something that tracks closely with Trump's professed "America first" and anti-"globalist" approach to foreign policy. In that vein Bolton led efforts to withdraw America's signature from the International Criminal Court.
But Bolton has also been a harsh critic of Russia -- a stance that might put him at odds with any effort to improve ties with Moscow, something Trump has stated as an objective of his administration.
Sen. Jeff Sessions, possible attorney general or secretary of defense
According to one source familiar with transition, the job of Defense Secretary is Sessions' if he wants it. Trump is also interested in the senator for attorney general, CNN's Dana Bash reports.
Sessions would also be a more conventional choice having represented Alabama in the Senate since 1997. He sits on the Senate Armed Services Committee and also served in the US Army Reserve from 1973-1986.
Sessions also meets the loyalty test having been the first sitting senator to endorse Trump during the campaign.
But Sessions is not a typical Washington insider, at times taking positions at odds with his colleagues on the Armed Services committee.
While Sessions has called for a defense spending boost, many of his other positions, such as a skeptical view of NATO and friendlier view of Russia, while at odds with his Republican senatorial colleagues brought him more into line with Trump's campaign comments.
Sessions shares Trump's hardline positions on immigration as well and is also under consideration for Attorney General.
Lt. Gen. Michael Flynn, possible national security adviser
On the defense side, retired Lt. Gen. Michael Flynn has emerged as a possible choice for national security adviser. Former military officers have found themselves in the role before, Barack Obama's first national security agency was former Gen. James Jones and Colin Powell served as the National Security Adviser to President Ronald Reagan.
Flynn, the former director of the Defense Intelligence Agency, is well versed in international and security affairs.
But Flynn, a vocal and sometimes fiery advocate for the president-elect during the campaign, has also echoed Trump's rhetoric of fighting terrorism, boosting ties with Russia, and casting a more critical eye on NATO.
Why there are so few conventional choices
A few more establishment type names are being floated for roles that oversee the intelligence community.
Burgess, the former DIA director, is also thought to be a potential selection for the director of national intelligence, which oversees the sprawling intelligence community.
And Pete Hoekstra, a long-serving former congressman from Michigan who chaired the House Intelligence Committee from 2004-2007 is thought to be in the running for the same role or director of the CIA.
While more conventional picks had initially been rumored for State and Defense such as the head of the Council of Foreign Relations, Richard Haas, and George W. Bush's National Security Adviser, Stephen Hadley, most analysts are beginning to believe that the odds favor a more conventional choice.
One of the reasons for the non-traditional short-list is that many of the Republican Party's most senior national security experts denounced their presidential candidate during the campaign as too reckless to lead the nation safely, somewhat limiting the pool from which to draw.
But some foreign policy luminaries have suggested that Republican experts should put aside their differences and Brent Scowcroft, George H.W. Bush's national security adviser national security adviser and who backed Hillary Clinton for president, told attendees as a ceremony at the Aspen Strategy Group Monday that "If you're asked to serve, please do. This man needs help."
One rumor has emerged that radio host Laura Ingraham could be Trump's White House spokesperson.
Asked outside of Trump Tower about that possibility, transition communications adviser Jason Miller did not address it specifically, but said to expect some interesting names.
"You're going to see a number of different names that are ultimately becoming a part of the President-elect's administration," Miller said. "There will be non-traditional names, a number of people who have had wide-ranging success in a number of different fields; wide-ranging success in business ... People will be excited when they see the type of leaders the President-elect brings into this administration."
This story has been updated.