The White House and the Pentagon declined to comment on Admiral Rogers’s fate. Reached by phone on Saturday afternoon, Admiral Rogers declined to comment.

The effort to force out Admiral Rogers, which was first reported by The Washington Post, puts Mr. Trump in the position of considering whether to name, as the man who would brief him on intelligence matters each morning, a four-star admiral whom the White House is considering relieving of his posts.

It also raises the question of why Mr. Obama would consider firing one of the nation’s top intelligence officers in the last days of his administration. Admiral Rogers’s replacement would not be confirmed until after Mr. Trump takes over. One senior intelligence official argued that letting word of the effort leak seemed more about politics or vengeance than about effecting any real change.

Representative Devin Nunes of California, the chairman of the House Intelligence Committee and a member of Mr. Trump’s transition team, strongly defended Admiral Rogers. In a letter to Mr. Carter and Mr. Clapper, Mr. Nunes asked them to testify before his committee to explain why they want to push Mr. Rogers out. “It’s not by accident that Admiral Rogers meets with the president-elect and two days later this story, which is completely built on lies, appears,” Mr. Nunes said in a short interview.

Mr. Carter and Mr. Clapper had submitted a formal recommendation to the White House to split the N.S.A., which conducts foreign surveillance and secures military networks, from the still-new Cyber Command. But there are questions inside the giant complex at Fort Meade, Md., where the N.S.A. and Cyber Command are housed, about whether the military cyberunit is ready to survive on its own. It relies heavily on the talent of the N.S.A., which dates back to the early 1950s.