The prospect of being the most powerful national security figure since Mr. Kissinger would hold obvious appeal for Mr. Pompeo. As President Richard M. Nixon’s national security adviser, Mr. Kissinger played an outsize role and effectively overshadowed Secretary of State William P. Rogers, so that when Mr. Rogers stepped down, it made sense to formalize his expanded role.

When Mr. Nixon resigned, President Gerald R. Ford kept Mr. Kissinger in both jobs, but ultimately the dual role came to be problematic and, in a broader reshuffling of his team, the president stripped Mr. Kissinger of his national security adviser title and left him at the State Department.

Mr. Trump told reporters on Thursday that he discussed a similar arrangement with Mr. Pompeo.

“I actually spoke to Mike Pompeo about that,” the president said. “I get along with him so well. We have a lot of the same views — a couple different views. But he likes the idea of having someone in there with him and I do too.” He added that he was considering 15 other candidates.

Putting Mr. Pompeo in dual roles like Mr. Kissinger would have been fraught with risks — some for Mr. Pompeo, and many for the national security establishment, which, in more normal times has come to view the National Security Council as a somewhat neutral arbiter among competing departments and agencies, from the Pentagon to the State Department to the intelligence agencies.

Colin H. Kahl, who was the top foreign policy aide for Vice President Joseph R. Biden Jr., noted that the national security adviser is expected to focus on the inside game, staffing the president and coordinating a collection of agencies and departments, while the secretary of state is the public face of American diplomacy.

“Given how much care and feeding Trump needs from staff, and how complex and fast moving the world is — even compared to Kissinger’s time — it is hard to imagine anyone effectively playing both roles,” Mr. Kahl said.

Yet Ms. Schake noted that Mr. Trump clearly does not want the kind of rigorous interagency process that other presidents have had and so in that sense there may be less of a problem in combining the roles. “But mainly what appointing Pompeo to both state and N.S.A. jobs would show is that, like Nixon, President Trump doesn’t actually trust anybody else,” she said.