Mr. Trump has ignored or downplayed Russian election interference since 2016, when the U.S. intelligence community agreed that Moscow had intervened to help elect him, a declaration that he feared would undermine his legitimacy. And the president has always been reluctant to shun or criticize groups that have supported him, including white supremacists and supporters of the QAnon conspiracy theory. That reluctance has dovetailed with his current desire to focus voter attention on unrest in cities that has been fomented by the Black Lives Matter movement.

And while Mr. Trump’s distaste for the intelligence agencies is not new, Mr. Murphy’s complaint provided some of the most graphic claims yet that career, nonpartisan analysts had been muzzled or shoved aside to downplay the threat posed specifically by Russia as it sought to sow discord and bolster Mr. Trump’s re-election campaign.

Mr. Murphy was demoted from his post in August to the Homeland Security Department’s management division after his office compiled intelligence reports on protesters and journalists in Portland, Ore. But he asserted in the complaint that his real offense was raising concerns to superiors about the directions he was given and for cooperating with the department’s inspector general. He asked the inspector general to investigate and reinstate him as the intelligence chief.

“Mr. Murphy followed proper, lawful whistle-blower rules in reporting serious allegations of misconduct against D.H.S. leadership, particularly involving political distortion of intelligence analysis and retaliation,” Mark S. Zaid, Mr. Murphy’s lawyer, said in a statement.

Democrats on the House Intelligence Committee asked Mr. Murphy to testify in private on Sept. 21, a possible precursor to a public hearing in the weeks before Election Day.

“We will get to the bottom of this, expose any and all misconduct or corruption to the American people, and put a stop to the politicization of intelligence,” said Representative Adam B. Schiff, Democrat of California and the chairman of the committee. He said the allegations of politically censored intelligence assessments were particularly worrisome in light of the Trump administration’s decision last month to stop briefing lawmakers in person on election security threats.

Sarah Matthews, a White House spokeswoman, said in a statement that Mr. O’Brien had “never sought to dictate the intelligence community’s focus on threats to the integrity of our elections or on any other topic.” She called Mr. Murphy a “disgruntled former employee” whom Mr. O’Brien had never heard of. But, she added, the national security adviser “consistently and publicly advocated for a holistic focus on all threats to our elections — whether from Russia, Iran, China or any other malign actor.”

Alexei Woltornist, the spokesman for the Homeland Security Department, rejected Mr. Murphy’s allegations.

“We flatly deny that there is any truth to the merits of Mr. Murphy’s claim,” Mr. Woltornist said. “D.H.S. looks forward to the results of any resulting investigation, and we expect it will conclude that no retaliatory action was taken against Mr. Murphy.”

As president, Mr. Trump has clashed frequently with the intelligence community, particularly over the issue of election interference. At a news conference in 2018 in Helsinki, Finland, Mr. Trump sided with President Vladimir V. Putin of Russia and said he did not think Moscow was responsible for the 2016 interference. A few months later, Mr. Trump dismissed the C.I.A.’s assessment that the Saudi crown prince was responsible for the murder of Jamal Khashoggi, a prominent Washington Post columnist. In February 2019, after intelligence chiefs offered Congress assessments on North Korea and Iran that were at odds with the White House, the president told them to “go back to school.”

Mr. Trump and his allies have also dismissed members of the intelligence community as overstating Russian espionage to undermine his presidency.

In his roles as a senior analyst and then the top intelligence official at the Homeland Security Department, Mr. Murphy was responsible for producing regular updates for the F.B.I., state election authorities and law enforcement on threats to U.S. elections by foreign powers.

In May, he said, Mr. Wolf directed him to stop providing assessments of the Russian threat, which favored Mr. Trump’s re-election, and instead focus on the activities of China and Iran, two other states that intelligence analysts have said pose a possible threat to the election and oppose Mr. Trump. China and Iran have so far not taken the kind of active measures that Moscow has, intelligence analysts say.

Mr. Murphy said he refused to comply, “as doing so would put the country in substantial and specific danger.”

Two months later, in July, Mr. Wolf again approached Mr. Murphy and asked him to hold back a warning that Russia was mounting a disinformation campaign around Mr. Biden’s mental health. That report also noted that China and Iran were criticizing Mr. Trump’s health.

“Mr. Cuccinelli stated that Mr. Murphy needed to specifically modify the section on white supremacy in a manner that made the threat appear less severe, as well as include information on the prominence of violent ‘left-wing’ groups,” according to the report. Mr. Murphy said that he had declined to make the modifications and that they constituted censorship.

Armed right-wing protesters outside the Oregon State Capitol in Salem, Ore., on Monday.Credit…Andrew Selsky/Associated Press

Mr. Murphy again pushed for the release of the report on July 8 in a meeting with Mr. Wolf and his deputy chief of staff, Scott Erickson, the complaint states. Mr. Wolf repeated Mr. Cuccinelli’s complaints and asked that the threat analysis include information on the unrest in Portland, Ore. Mr. Murphy said he did not agree with edits that undermined the gathered intelligence.

Mr. Murphy said in the report that the final version of the assessment would be altered to emphasize “anarchist” groups and antifa, a loose collection of left-wing anti-fascist activists that former Homeland Security analysts had determined to be not as deadly as more organized white nationalists.

During meetings in May and July, Mr. Cuccinelli and Mr. Wolf directed Mr. Murphy to change other intelligence briefings that were shared with federal and state law enforcement agencies to ensure that they matched Mr. Trump’s descriptions of the danger posed by antifa.

The allegations from Mr. Murphy are only the latest to hit the Homeland Security Department, which was originally built to protect the United States from foreign terrorism and improve responses to national emergencies. Mr. Wolf’s deployment of teams of tactical agents to Portland prompted widespread criticism from former Homeland Security officials and members of Congress that the agency, which is also responsible for carrying out most of Mr. Trump’s immigration policies, had become a tool for the president’s re-election campaign.

But Mr. Murphy has also been the subject of scrutiny at both the Homeland Security Department and the F.B.I.

He was reassigned by Mr. Wolf after his office distributed intelligence bulletins that partly targeted The New York Times for publishing an analysis indicating that the Homeland Security Department had little understanding of the situation in Portland when it deployed teams of tactical agents in camouflage to face crowds of protesters. Mr. Murphy said in the complaint that his office did not conduct surveillance of reporters’ private data but rather included their social media posts in the bulletins.

The analysis also summarized the tweets of Benjamin Wittes, the editor in chief of the blog Lawfare, including one that had showed an email from Mr. Murphy telling intelligence officers to refer to individuals attacking the federal courthouse in Portland as “VIOLENT ANTIFA ANARCHISTS.”

“It is certainly possible that Murphy could have been a bad actor in some respects, which I think he clearly was, but also put out legitimate claims as a whistle-blower,” Mr. Wittes said. “It’s also possible that he is protecting himself by making allegations.”

Hours before the release of Mr. Murphy’s complaint, Mr. Wolf gave an annual address to summarize the work of the Homeland Security Department that was emblematic of the agency’s changing priorities under the Trump administration.

He blamed China and the World Health Organization for the coronavirus pandemic and local city and state officials for the unrest in Portland, while promoting the construction of Mr. Trump’s wall along the southwest border. Mr. Wolf also touched on the department’s efforts to counter domestic extremism.

“Let me be clear: D.H.S. stands in absolute opposition to any form of violent extremism, whether by white supremacists or anarchist extremists,” he told the audience.

Mr. Wolf added that the agency would issue the delayed plan for preventing extremist violence this week, although the department did not respond to inquiries of when it would be made public.