Still, in interviews with nearly a dozen of Mr. Ayers’s confidants, White House officials and former administration officials, Mr. Ayers was invariably described as someone who could bring to the White House the political skills that Mr. Kelly lacks. And, they say, he has been effective in protecting and elevating Mr. Pence while also forging alliances with influential members of Mr. Trump’s inner circle, including his son-in-law, Jared Kushner, and daughter Ivanka Trump.

But the president tends to keep staffing decisions close to his chest, often announcing his choices after lengthy cliffhangers. One senior administration official said Mr. Ayers is not the only possible replacement Mr. Trump has considered.

Mr. Ayers did not respond to a request for comment for this article, nor did the White House or the vice president’s office.

But last month, Mr. Trump sought to tamp down rumors by inviting a reporter for New York magazine into the Oval Office, where Mr. Kelly, Mr. Ayers and Mr. Pence were ready to display their affection for each other.

Privately, Mr. Ayers has done little to stop the speculation. According to a former senior administration official, he has openly discussed with his allies the possibility of succeeding Mr. Kelly, something that has not gone unnoticed in the West Wing.

Even if he does not end up next to Mr. Trump in the Oval Office, it has been a rapid and lucrative rise for Mr. Ayers since his start as a 20-year-old personal aide in Mr. Perdue’s come-from-behind-victory as Georgia’s first Republican governor since Reconstruction.

Less than two years later, Mr. Ayers was named campaign manager for Mr. Perdue’s successful re-election campaign, and after that, he took a job running the Republican Governors Association. At the time, the group was something of an afterthought in the Republican big-money landscape. But Mr. Ayers grew it into a political juggernaut that corralled six- and seven-figure donations from some of the biggest donors on the right.