A top House Democrat accused the administration Thursday of stonewalling a probe into a national security whistleblower’s complaint against President Trump by raising executive privilege, but Presidents Bill Clinton and Barack Obama also insisted on the right to privilege in such cases.

Emerging from a three-hour, closed-door briefing with an inspector general, Rep. Adam B. Schiff, California Democrat and chairman of the House Permanent Select Committee on Intelligence, said the Justice Department is shielding the White House by barring disclosure of details about the whistleblower’s complaint.

“This shows someone is trying to manipulate the system to keep information about an urgent matter from the Congress,” Mr. Schiff told reporters. “There is no privilege that covers whether the White House is involved and trying to stifle a whistleblower complaint. There’s no privilege to be corrupt.”

Mr. Schiff is demanding that the administration turn over the complaint, which reportedly says the president made an unspecified “promise” to an unspecified foreign head of state during a phone conversation sometime this summer. Intelligence Community Inspector General Michael Atkinson deemed the complaint credible and of “urgent concern,” turning heads on Capitol Hill.

Mr. Trump on Thursday rejected accusations that he made an inappropriate promise to a foreign leader. He said he wouldn’t be “dumb enough” to take such action while other officials were monitoring his phone conversations.

“Virtually anytime I speak on the phone to a foreign leader, I understand that there may be many people listening from various U.S. agencies, not to mention those from the other country itself. No problem!” the president tweeted.

Mr. Trump added, “Knowing all of this, is anybody dumb enough to believe that I would say something inappropriate with a foreign leader while on such a potentially ‘heavily populated’ call. I would only do what is right anyway, and only do good for the USA!”

The president characterized the report by The Washington Post as “another Fake News story out there — It never ends!”

Mr. Atkinson told the committee that the whistleblower’s complaint was related to multiple acts, not one phone conversation, lawmakers said, but he wouldn’t discuss specifics or confirm that the complaint targeted the president.

“He didn’t talk about anything about the allegations, where he was very protective,” said Rep. Mike Quigley, Illinois Democrat. “But he did mention that this complaint was based on a series of events, ‘more than one,’ to get the exact wordage right.”

Rep. Val Butler Demings, Florida Democrat, said, “We heard the word ‘privilege’ a lot this week.”

“His answer centered around the word ‘privilege,’ that it might be privileged information,” Ms. Demings said of Mr. Atkinson on CNN. “While he did not say it involved the administration, it certainly led me to believe that his instructions may have been coming from the administration.”

But she disputed the characterization of multiple events, saying Mr. Atkinson wouldn’t discuss specifics.

Some analysts say there are legitimate questions about the balance of power and what acting Director of National Intelligence Joseph Maguire is required to disclose to Congress, given presidential authority.

Robert Litt, who served as general counsel to the Office of the Director of National Intelligence in the Obama administration, noted that Mr. Clinton asserted the right to executive privilege when he signed into law the Intelligence Community Whistleblower Protection Act of 1998.

“The executive branch has always maintained that it does not consider the statutory language mandatory,” Mr. Litt wrote on the Lawfare blog. “In signing the original Intelligence Community Whistleblower Protection Act of 1998, President Clinton stated that it ‘does not constrain my constitutional authority to review and, if appropriate, control disclosure of certain classified information to Congress.’

“President Obama reiterated this limitation in 2010,” he said. “Congress no doubt disagrees with this interpretation, but the president’s ultimate control over classified information has been a consistent position of every administration.”

Jack Goldsmith, a Harvard Law professor and senior fellow at the Hoover Institution, said the president’s power to act in confidence “is at its absolute height when he has a classified conversation with a foreign leader.”

A president “cannot conduct foreign policy if his or her controversial secret foreign policy communications can be disclosed at the determination of an intelligence employee,” Mr. Goldsmith tweeted.

“Putting it brutally, Article II gives the president the authority to do, and say, and pledge, awful things in the secret conduct of U.S. foreign policy,” he said in a Twitter post. “That is a very dangerous discretion, to be sure, but has long been thought worth it on balance.”

The whistleblower is being represented by Andrew P. Bakaj, a former CIA officer and managing partner at the Compass Rose Legal Group, a national security law firm.

Mr. Atkinson has agreed to appear in an open session with lawmakers Thursday.

Mr. Schiff is scolding Mr. Maguire for refusing to share details of the complaint, which was filed Aug. 12, saying he is required to notify committees because of the inspector general’s determination.

“Given the inspector general said, ‘This is urgent,’ it can’t wait,” Mr. Schiff said. “Even if you could make a colorable claim of privilege over the subject matter of the complaint, given that it involves something that the IG has already found to be serious and credible and evidence of wrongdoing on one kind or another, there is no privilege that covers that. There’s no privilege to conceal that. We can’t get an answer because the Department of Justice and the director of national intelligence will not authorize the IG to tell us.”

Rep. Jackie Speier, a California Democrat who attended the briefing, called the administration’s actions “deeply troubling” and “a violation of the law.”

Mr. Schiff indicated that the spy community’s inspector general would not reveal to lawmakers whether the president is the target of the complaint.

“I don’t know whether the White House is directly involved because we can’t get an answer to that question, but we do know that they are making some claim that a privilege may apply,” he told reporters. “Well, that narrows the categories who may be intervening here.”

Former FBI and CIA official Phil Mudd said on CNN that the whistleblower had no right to raise concerns about Mr. Trump’s calls with foreign leaders.

“The president can say what he wants,” Mr. Mudd said. “It’s not the responsibility of the intel guys to go police the president and go snitch on him to the Congress. Ridiculous.”

Some allies of the president said the complaint was another example of the federal bureaucracy and the intelligence community’s long-running opposition to the president. Judicial Watch President Tom Fitton called it “another Deep State attack” on Mr. Trump.

Democrats and their pro-impeachment allies said the developments are more troublesome for Mr. Trump than the now-discredited Russia investigation. Former Nixon White House Counsel John Dean said Attorney General William P. Barr “is protecting Trump from a whistleblower complaint.”

“This may take Impeachment in a totally new direction,” he tweeted.

Obama administration acting Solicitor General Neal Katyal said the Justice Department and the administration “are contorting themselves backwards to try to hide this.”

“There are probably tapes and transcripts documenting a gross abuse of power by Trump,” Mr. Katyal tweeted. “Gonna be ugly.”

• Tom Howell Jr. contributed to this report.

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