Ms. Gordon, who has served more than 30 years in intelligence posts at the C.I.A. and other agencies, has not been officially informed by the White House that Mr. Trump intends to name someone else to oversee the intelligence agency until the Senate confirms a new director of national intelligence, officials said.

But the White House requested this week that the office provide a list of senior officials who worked for the agency, according to a senior administration official — a move that was interpreted as another sign that it is looking beyond her for people who could be temporarily installed in the top position.

When Mr. Trump posted tweets Sunday announcing that Mr. Coats would step down on Aug. 15 and that he intended to nominate Mr. Ratcliffe, the president hinted that Ms. Gordon might not automatically become the acting director in the interim, saying an acting director would be named soon.

Those tweets prompted concern on Capitol Hill that Mr. Trump would circumvent Ms. Gordon. The next day, Senator Richard M. Burr, Republican of North Carolina, who leads the Senate Intelligence Committee, expressly referred to the fact that he looked forward to working with Ms. Gordon, calling her “a trusted partner.”

On Friday, Senator Mark Warner, Democrat of Virginia, who is the committee’s vice chairman, said that the law was “quite clear” that the acting role goes to the deputy when the director of national intelligence leaves and that Ms. Gordon had the Senate’s confidence. “It’s outrageous if the president is hoping to pass over this extremely qualified and experienced individual, the highest-ranking woman in O.D.N.I., in order to install a political loyalist as acting director,” he said.

Ms. Gordon’s experience is not necessarily a point in her favor for the White House, where Mr. Trump and his allies view the permanent bureaucracy of national security professionals with suspicion as a so-called deep state that may be out to get him.

Mr. Trump and House Republicans have made clear that they believe a broad reorganization of the Office of the Director of National Intelligence is needed. Administration officials and House Republicans also have said they would like someone at the agency who will work well with Attorney General William P. Barr, who has ordered a review of the intelligence agencies’ support for the F.B.I. as the bureau sought to understand Moscow’s covert efforts to tilt the 2016 election, including any links to the Trump campaign.