The White House chief of staff is typically tasked in large part with ensuring that all wheels are spinning in the complex White House organization, and the source said that some people in Trump's orbit do not think Bannon, the executive chairman of Breitbart News who joined Trump's campaign in August, is the best fit for that position.
Trump's contemplation of Bannon as chief of staff comes as his presidential transition team is feverishly ramping up its efforts to build out an administration after his surprising win Tuesday.
It also highlights how Trump is contending with competing impulses as to whether he should fill the ranks of his White House with the right-wing figures who energized his campaign or more establishment figures experienced in the ways of Washington.
Reince Priebus, the RNC chairman who has become a close confidante of Trump's in the final months of the campaign, is also reportedly in the running for the top White House post.
Priebus is an experienced Washington politico with establishment credo who has also earned Trump's trust as he put the weight of the RNC's ground and data operation behind Trump's candidacy, and stuck by Trump's side through the controversies that dogged his campaign.
Priebus and Trump have talked about the chief of staff position, according to a source with knowledge of the transition process who added that "I think he (Priebus) really wants it."
The New York Times
first reported Thursday that Bannon was being considered for the post, though the has previously said he was not interested in joining the administration should Trump be elected.
Both Priebus and Bannon have traveled with Trump in the final months of his campaign. Bannon in the final week of the campaign was frequently spotted in the wings at Trump rallies at the side of Jared Kushner, Trump's son-in-law who has helped manage the presidential campaign.
Bannon has also been a major force behind some of Trump's more controversial stunts, including when Trump held an impromptu press event with women who had accused former President Bill Clinton of sexual assault and misconduct. Bannon was spotted in the back of the room smiling as reporters were led in for the debate night surprise.
Scrambling to fill jobs; Trump to meet with transition team
The President-elect's transition team is rushing to fill top cabinet positions as well as the hundreds of posts that require a security clearance.
A transition source told CNN on Thursday that Trump's transition team will brief the newly-elected president on their planning for the transition to a new administration.
As those staffers work to vet the individuals who will serve in the 15 executive departments, they are also finding themselves competing with lobbying firms, which are scrambling to hire Republicans in Washington who can do business with the new administration.
Many of those K Street firms have spent months hiring more Democrats with the assumption in mind that Clinton would win the election.
That adds to the Trump transition team's already more limited pool of qualified contenders as they face a long list of Republicans who have rebuked Trump.
A GOP official who is helping to find candidates for some of the 4,000 government jobs the transition will eventually have to fill said the work is made harder by the competition from K Street, which is also working to scoop up 30-something staffers on Capitol Hill who are a prime target for transition efforts.
Adding to that, there are still some young staffers who appear to be resisting opportunities to work in a Trump administration for fear it will hurt their careers.
No contact with key agencies
The Trump transition team still remains in the early stages of its efforts and has yet to reach out to some key executive departments, including the State Department, Pentagon and Justice Department.
Pentagon press secretary Peter Cook said Thursday that while the department stands "ready to engage with the designated transition leaders...to the best of my knowledge that has not yet happened."
The State Department's deputy spokesman Mark Toner also said the Trump transition team has not yet put a single envoy in touch with the diplomatic agency to discuss transition plans. And while the Justice Department expected Trump envoys on Wednesday, officials from the transition team are now expected to meet with officials at that department on Monday.