The two White House advisers insisted, however, that Mr. Tillerson is expected to leave sometime in January.

In the plan revealed on Thursday, Mr. Tillerson would be succeeded by Mike Pompeo, currently the C.I.A. director, who is more closely aligned with Mr. Trump on a series of important foreign policy matters. Mr. Tillerson’s refusal to resign despite his disagreements with Mr. Trump — and the president’s reported dislike for the diplomat — may have been the reason senior administration officials leaked Mr. Kelly’s plan this week to replace him.

Mr. Tillerson had previously and repeatedly denied reports of his impending departure, even going so far as to hold a news conference in October to say, “There’s never been a consideration in my mind to leave.”

The debate over Mr. Tillerson’s status is the latest outgrowth of a White House that has been beset by internal battles, high-level departures, indictments of two top former campaign officials and guilty pleas from a former national security adviser and a campaign adviser. All have contributed to a generalized sense of chaos that shows few signs of abating, even as the administration nears the completion of its first year in power.

Mr. Tillerson is a phlegmatic former chief executive of Exxon Mobil whose personal net worth is estimated in the hundreds of millions of dollars. His continued service to a White House that so obviously wants him gone has led to something of a cottage industry of speculation in Washington about his reasons for remaining.

One theory, frequently repeated at the State Department, is that an early departure would cost Mr. Tillerson millions of dollars in taxes for assets he sold to take the diplomatic post. Tax experts have dismissed this notion.

After being nominated, Mr. Tillerson was granted a so-called Certificate of Divestiture that allowed him to sell off Exxon stock without having to immediately pay capital gains taxes on those holdings. Two lawyers involved in issuing these documents said that the federal law creating this process — which is intended to eliminate a disincentive to qualified individuals joining the government — has no provision to withdraw this benefit, assuming that Mr. Tillerson completed the sale of his stocks before he left his government job.