Against this backdrop, Mr. Trump has appointed Mike Pompeo, a former Republican congressman from Kansas, to run the C.I.A., and former Senator Dan Coats, an Indiana Republican, to be the director of national intelligence (he is still awaiting confirmation). Both were the preferred choices of the Republican congressional leadership and Vice President Mike Pence and had no close or longstanding ties to Mr. Trump. In fact, they each endorsed Senator Marco Rubio of Florida for president during the 2016 Republican primaries.

But the potential White House role for Mr. Feinberg follows intense speculation among intelligence professionals that Mr. Feinberg is in line for a powerful position within the intelligence community.

Reports that Mr. Feinberg was under consideration to run the clandestine service rocked the intelligence community in recent weeks, raising the prospect of direct White House control over America’s spies at a time when Mr. Trump’s ties to Mr. Putin are under investigation by the F.B.I. and congressional committees.

The last time an outsider with no intelligence experience took the job was in the early days of the Reagan administration, when Max Hugel, a businessman who had worked on Mr. Reagan’s campaign, was named to run the spy service. His tenure at the C.I.A. was marked by turmoil and questions about the politicization of the agency. He was forced to resign after six months, amid accusations about his past business dealings. (He later won a libel case against the two brothers who made the accusations.)

Even the prospect that Mr. Feinberg may lead a review for the White House has raised concerns in the intelligence community.

Mr. Coats is especially angry at what he sees as a move by Mr. Bannon and Mr. Kushner to sideline him before he is even confirmed, according to current and former officials. He believes the review would impinge on a central part of his role as the director of national intelligence and fears that if Mr. Feinberg were working at the White House, he could quickly become a dominant voice on intelligence matters.