WASHINGTON — Senior officials across the government became convinced in January that the incoming national security adviser, Michael T. Flynn, had become vulnerable to Russian blackmail.

At the F.B.I., the C.I.A., the Justice Department and the Office of the Director of National Intelligence — agencies responsible for keeping American secrets safe from foreign spies — career officials agreed that Mr. Flynn represented an urgent problem.

Yet nearly every day for three weeks, the new C.I.A. director, Mike Pompeo, sat in the Oval Office and briefed President Trump on the nation’s most sensitive intelligence — with Mr. Flynn listening. Mr. Pompeo has not said whether C.I.A. officials left him in the dark about their views of Mr. Flynn, but one administration official said Mr. Pompeo did not share any concerns about Mr. Flynn with the president.

The episode highlights a remarkable aspect of Mr. Flynn’s tumultuous, 25-day tenure in the White House: He sat atop a national security apparatus that churned ahead despite its own conclusion that he was at risk of being compromised by a hostile foreign power.

The concerns about Mr. Flynn’s vulnerabilities, born from misleading statements he made to White House officials about his conversations with the Russian ambassador, are at the heart of a legal and political storm that has engulfed the Trump administration. Many of Mr. Trump’s political problems, including the appointment of a special counsel and the controversy over the firing of the F.B.I. director, James B. Comey, can ultimately be traced to Mr. Flynn’s stormy tenure.

Time and again, the Trump administration looked the other way in the face of warning signs about Mr. Flynn. Mr. Trump entrusted him with the nation’s secrets despite knowing that he faced a Justice Department investigation over his undisclosed foreign lobbying. Even a personal warning from President Barack Obama did not dissuade him.

Mr. Pompeo sidestepped questions from senators last month about his handling of the information about Mr. Flynn, declining to say whether he knew about his own agency’s concerns. “I can’t answer yes or no,” he said. “I regret that I’m unable to do so.” His words frustrated Senator Ron Wyden, an Oregon Democrat and a member of the Senate Intelligence Committee.

“Either Director Pompeo had no idea what people in the C.I.A. reportedly knew about Michael Flynn, or he knew about the Justice Department’s concerns and continued to discuss America’s secrets with a man vulnerable to blackmail,” Mr. Wyden said in a statement. “I believe Director Pompeo owes the public an explanation.”

After Mr. Pompeo’s Senate testimony, The New York Times asked officials at several agencies whether Mr. Pompeo had raised concerns about Mr. Flynn to the president and, if so, whether the president had ignored him. One administration official responded on the condition of anonymity that Mr. Pompeo, whether he knew of the concerns or not, had not told the president about them.

A C.I.A. spokesman declined to discuss any interactions between the president and Mr. Pompeo.

“Whether the C.I.A. director briefed the president on a specific intelligence issue during a specific time frame is not something we publicly comment on, and we’re not about to start today,” said the spokesman, Dean Boyd.

Concerns across the government about Mr. Flynn were so great after Mr. Trump took office that six days after the inauguration, on Jan. 26, the acting attorney general, Sally Q. Yates, warned the White Housethat Mr. Flynn had been “compromised.”

Ms. Yates’s concerns focused on phone calls that Mr. Flynn had in late December with Sergey I. Kislyak, the Russian ambassador to the United States. When the White House faced questions about whether the two men had discussed lifting American sanctions on Russia, Vice President Mike Pence told reporters that Mr. Flynn had assured him that sanctions were not discussed. Intelligence officials knew otherwise, based on routine intercepts of Mr. Kislyak’s conversations.

“That created a compromise situation,” Ms. Yates later told Congress, “a situation where the national security adviser essentially could be blackmailed by the Russians.”

Mr. Trump waited 18 days from that warning before firing Mr. Flynn, a period in which Mr. Pompeo continued to brief Mr. Flynn and the president. The White House has offered changing explanations for why the president waited until Feb. 13 — soon after Ms. Yates’s warning made national news — before firing Mr. Flynn.

White House officials have said they moved deliberately both out of respect for Mr. Flynn and because they were not sure how seriously they should take the concerns. They also said the president believed that Ms. Yates, an Obama administration holdover, had a political agenda. She was fired days later over her refusal to defend in court Mr. Trump’s ban on travel for people from several predominantly Muslim countries.

A warning from Mr. Pompeo might have persuaded the White House to take Ms. Yates’s concerns more seriously. Mr. Pompeo, a former congressman, is a Republican stalwart whom Mr. Trump has described as “brilliant and unrelenting.”

Mr. Pompeo was sworn in three days before Ms. Yates went to the White House. He testified last month that he did not know what was said in that meeting. By that time, C.I.A. officials had attended meetings with F.B.I. agents about Mr. Flynn and reviewed the transcripts of his conversations with the Russian ambassador, according to several current and former American security officials. Separately, intelligence agencies were aware that Russian operatives had discussed ways to use their relationship with Mr. Flynn to influence Mr. Trump.

Mr. Pompeo, who briefs the president nearly every day, had frequent opportunities to raise the issue with Mr. Trump.

The President’s Daily Brief is a rundown of what America’s spies consider the most pressing issues facing the United States. On any given day, it can include details of a terrorist plot being hatched overseas, an analysis of a foreign political crisis that threatens American interests or a look at foreign hackers who are trying to breach American government computer systems.

Each president takes the briefing differently. Mr. Obama was said to prefer reading it on a secure tablet. President George W. Bush liked his briefers to talk through the document they were presenting. Mr. Pompeo has described Mr. Trump as a voracious consumer of the briefing who likes maps, charts, pictures, videos and “killer graphics.”

At an event last month at Westwood Country Club in Northern Virginia, Mr. Pompeo told retired C.I.A. officials that his briefings often ran past their scheduled 30 minutes, according to one retired official in attendance. Mr. Pompeo said Mr. Trump was eager for information and asked many questions.

At his confirmation hearing, Mr. Pompeo assured senators that he would provide the president with unvarnished information, even when it would be viewed as unpleasant. “I can tell you that I have assured the president-elect that I’ll do that,” Mr. Pompeo said.

On Capitol Hill, Mr. Wyden questioned why Mr. Pompeo continued having discussions with Mr. Flynn despite the concerns of intelligence officials. “He was the national security adviser,” Mr. Pompeo said. “He was present for the daily brief on many occasions.”

Mr. Flynn had no love for the C.I.A., and the feeling was mutual. An Army general who had risen to lead the Defense Intelligence Agency, Mr. Flynn emerged in retirement as a C.I.A. critic, blaming the agency for his firing and what he called its failure to foresee the rise of the Islamic State. He insisted the Obama administration had politicized the agency, an assertion Mr. Pompeo later said he saw no evidence to support.