“We’re up against a permanent bureaucratic structure defending itself and quite willing to break the law to do so,” Mr. Gingrich said.

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Neither Mr. Trump nor Mr. Bannon has used the term “deep state” publicly. But each has argued that there is an orchestrated effort underway, fueled by leaks and enabled by the news media, to cut down the new president and interfere with his agenda.

“Reports concerning potentially politically motivated investigations immediately ahead of the 2016 election are very troubling,” Sean Spicer, the White House press secretary, said on Sunday.

Mr. Bannon, speaking last month at the Conservative Political Action Conference, said a central element of Mr. Trump’s presidency would be the “deconstruction of the administrative state.” It was his latest articulation of a dim view of federal agencies that he argued had grabbed power at the behest of the “progressive left.”

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Breitbart News, the conservative site Mr. Bannon used to run, uses the term deep state frequently in its coverage, including in a story on Sunday headlined “DeepStateGate: Trump Ends the Wiretapping Innuendo Game by Dealing Himself In.” The term has gained currency on other right-leaning websites, conservative talk radio and on social media, where Mr. Trump’s supporters are inflamed by the notion that a powerful secret cabal is plotting his downfall.

Projecting a new role for Obama

Veterans of prior administrations have been alarmed by the charge, arguing that it suggests an undemocratic nation where legal and moral norms are ignored.

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“ ‘Deep state’ I would never use,” Michael V. Hayden, the former Central Intelligence Agency director under both Mr. Obama and former President George Bush, said on MSNBC on Monday. “That’s a phrase we’ve used for Turkey and other countries like that, but not the American republic.”

Loren DeJonge Schulman, a former top official in Mr. Obama’s National Security Council who is a senior fellow at the Center for a New American Security, said presidents and top White House officials often bristle at what they consider to be a sluggish bureaucracy. But it is jarring for an administration in power to claim that civil servants are actively working to subvert the government.

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“A deep state, when you’re talking about Turkey or Egypt or other countries, that’s part of government or people outside of government that are literally controlling the direction of the country no matter who’s actually in charge, and probably engaging in murder and other corrupt practices,” Ms. Schulman said. “It’s shocking to hear that kind of thinking from a president or the people closest to him.”

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Yet to Mr. Trump’s allies and supporters, the president is giving voice to a favorite theory.

“We are talking about the emergence of a deep state led by Barack Obama, and that is something that we should prevent,” said Representative Steve King, Republican of Iowa. “The person who understands this best is Steve Bannon, and I would think that he’s advocating to make some moves to fix it.”

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Mr. King cited as evidence of a thriving deep state Mr. Obama’s decision to stay in Washington after leaving the White House, a decision he said was driven by the former president’s desire to frustrate Mr. Trump’s agenda. (Mr. Obama has said he is remaining in Washington until his younger daughter, Sasha, graduates from high school in 2019.)

Mr. Trump “needs to purge the leftists within the administration that are holdovers from the Obama administration, because it appears that they are undermining his administration and his chances of success,” Mr. King said.

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Pakistan, home to military coups, is considered Exhibit A

The deep state is a phrase often heard in countries where there is a history of military coups and where generals often hold power independent of elected leaders.

Pakistan is Exhibit A: The deep state is often invoked in serious discussions about the role of the Pakistani military and its intelligence service, the Directorate of Inter-Services Intelligence.

Wide swaths of the population see the unseen hand of the security services behind major political events and all kinds of everyday happenings, such as random traffic stops.

The views are not without basis — there have been repeated military coups in Pakistan, and the military and the spy service often operate largely independent of the country’s politicians.

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“The deep state concept emerges in places where the army and the security apparatus creates boundaries within which the civilian political people are allowed to operate,” said Peter Feaver, a specialist in civil-military issues at Duke University and a national security aide to Mr. Bush. “If they transgress those boundaries, then the deep state interferes to reorder things, often using military force.”

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Leaks vs. serious opposition

“There are milder forms of it in healthier democracies,” Mr. Feaver said, arguing that American presidents have often chafed against the constraints of the federal bureaucracy.

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“Nixon shared a similar kind of distrust of the government and felt the government was out to get him at points,” Mr. Feaver added. “President Trump’s view seems to be more on the Nixon part of the spectrum, which is far from the Pakistan part.”

In the United States, it is hardly unusual for dissent among warring factions inside the government to burst into public view. Under former President Ronald Reagan, the secretary of state, George P. Shultz, and the secretary of defense, Caspar W. Weinberger, were often at odds and would feud through dueling news reports.

“Just because you see things like leaks and interference and obstruction doesn’t necessarily mean there’s a deep state — that’s something we’ve seen before, historically, and it’s nothing new,” said James Jay Carafano, a fellow at the Heritage Foundation who advised Mr. Trump’s transition. “What would be different is if there were folks from the previous administration that were consciously orchestrating, in a serious way, inside opposition to the president.”

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In the absence of evidence one way or the other, Mr. Carafano added, “It’s hard to know: Is this Trump using some strong political rhetoric, or an actual theory?”