At the same time, there was some concern in the White House about the appearance of a rush to the exits given that other senior officials may also leave in the early part of the new year. White House officials were debating whether it would be better to spread out any departures or just get them over with all at once.
The ouster of Mr. Tillerson would end a turbulent reign at the State Department for the former Exxon Mobil chief executive, who has been largely marginalized over the last year. Mr. Trump and Mr. Tillerson have been at odds over a host of major issues, including the Iran nuclear deal, the confrontation with North Korea and a clash between Arab allies. The secretary was reported to have privately called Mr. Trump a “moron” and the president publicly criticized Mr. Tillerson for “wasting his time” with a diplomatic outreach to North Korea.
Mr. Tillerson’s departure has been widely anticipated for months, but associates have said he was intent on finishing out the year to retain whatever dignity he could. Even so, an end-of-year exit would make his time in office the shortest of any secretary of state whose tenure was not ended by a change in presidents in nearly 120 years.
While some administration officials initially expected him to be replaced by Nikki R. Haley, the ambassador to the United Nations, Mr. Pompeo has become the White House favorite.
Mr. Pompeo, a former three-term member of Congress, has impressed Mr. Trump during daily intelligence briefings and become a trusted policy adviser even on issues far beyond the C.I.A.’s normal mandate, like health care. But he has been criticized by intelligence officers for being too political in his job.
Mr. Cotton has been perhaps Mr. Trump’s most important supporter in the Senate on national security and immigration and a valued outside adviser. Officials cautioned that there was still a debate about whether Mr. Cotton was more valuable to the president in the Senate than in taking over the spy agency in Langley, Va., but he is the consensus choice at the moment.
Under Arkansas state law, Gov. Asa Hutchinson, a Republican, would appoint a replacement who could serve until the 2018 election. That could put another seat in play during a midterm election when Republicans, with 52 of 100 seats in the Senate, cannot afford to take too many chances. If Mr. Cotton stayed in the Senate, his seat would not be up for election again until 2020.
Asked about a possible move, Caroline Rabbitt Tabler, a spokeswoman for Mr. Cotton, said, “Senator Cotton’s focus is on serving Arkansans in the Senate.”
Mr. Tillerson’s appointment was something of an experiment from the start. Never before had a president named a secretary of state with no prior experience in government, politics or the military. Mr. Trump, who himself had no government or military experience before this year, bet that Mr. Tillerson would be able to translate his formidable skills in the corporate world to international diplomacy after 41 years at Exxon Mobil.
But Mr. Tillerson has often been on a different page than Mr. Trump, and he has spent much of his time reorganizing the State Department, slashing its budget and pushing out more than 2,000 career diplomats. Even on that he ran into serious troubles. Just this week, the counselor he brought in to execute his plan quit after just three months.
A sign of his fading fortunes in the White House has been the changing views of Mr. Kelly. Once a defender of Mr. Tillerson, Mr. Kelly is described by colleagues as now having mixed opinions, seeing him as a wounded figure who may no longer be able to be as effective as the president needs his secretary to be.