Two men masquerading as Department of Homeland Security officers in Washington duped several members of the Secret Service, providing them with tens of thousands of dollars in gifts, including rent-free apartments, federal prosecutors said. One of the men told several people that he had connections to Pakistan’s main intelligence agency, prosecutors said in federal court on Thursday.

Four members of the Secret Service, the agency charged with protecting the president and the president’s family, have been placed on administrative leave while the case is being investigated, according to an affidavit filed in federal court in the District of Columbia.

One of the men, Arian Taherzadeh, offered to buy a $2,000 assault rifle for an agent assigned to Jill Biden’s protective detail, according to the affidavit, which was filed Tuesday.

Mr. Taherzadeh, 40, and Haider Ali, 35, were charged with one count of false impersonation of an officer of the United States, the affidavit said.

Both men appeared by videoconference on Thursday at a hearing in federal court in Washington, where prosecutors said that Mr. Ali had told witnesses that he had connections to the Inter-Services Intelligence in Pakistan. Mr. Ali also held several visas that had been issued by Pakistan and Iran, prosecutors said.

The F.B.I. arrested the men, both of whom are U.S. citizens, on Wednesday in Southeast Washington. The two-year scheme began in February 2020 and unraveled recently, the affidavit said.

“Taherzadeh and Ali have attempted to use their false and fraudulent affiliation with D.H.S. to ingratiate themselves with members of federal law enforcement and the defense community,” David Elias, an F.B.I. special agent, wrote in the affidavit.

Mr. Elias did not say why the men carried out the impersonation. Federal prosecutors declined to give a motive or provide further information about the two men.

Joshua Rothstein, an assistant U.S. attorney, told the judge during the court proceeding on Thursday that both men were flight risks and should remain in custody. A detention hearing for both defendants, who did not enter pleas, is scheduled for Friday afternoon.

Mr. Rothstein said that investigators had searched five apartments and several cars that Mr. Taherzadeh and Mr. Ali had used. He said they found equipment and paraphernalia usually associated with law enforcement, including body armor, gas masks, zip ties, hand-held radios, a drone like the ones used by S.W.A.T. teams, Homeland Security patches, 40 to 50 rounds of ammunition, weapon stocks and documents that were stamped “law enforcement sensitive.”

Mr. Taherzadeh told Magistrate Judge G. Michael Harvey in court on Thursday that he had completed “some college” at Georgetown University. Michelle Peterson, a federal public defender who represented both men, said they planned to ask to be released at the hearing on Friday.

If convicted on the impersonation charge, they face up to three years in prison and a $250,000 fine.

According to the affidavit, Mr. Taherzadeh provided members of the Secret Service and a Homeland Security employee rent-free apartments, iPhones, surveillance systems, a drone, a flat-screen television, a generator and case to store an assault rifle.

Mr. Taherzadeh also offered them what he described as “official government vehicles.”

The White House did not immediately respond to a request for comment on Thursday. Michael LaRosa, a spokesman for Dr. Biden, and the Homeland Security Department referred questions to the Secret Service.

In a statement, a spokesman for the Secret Service said that the agency was working with investigators and that the employees involved in the case had been restricted “from accessing Secret Service facilities, equipment, and systems.”

“The Secret Service adheres to the highest levels of professional standards and conduct and will remain in active coordination with the Departments of Justice and Homeland Security,” the spokesman, Kang Lee, said.

In a statement, the Pakistani embassy said on Friday: “Several media outlets have carried a news story mentioning that one individual, named Haider Ali, who was arrested on the charges of impersonating as federal agent in Washington, D.C., had claimed that he was affiliated with Pakistan’s Inter-Services Intelligence. The claim of Mr. Haider Ali is totally fallacious. The Embassy categorically rejects this false claim.”

It was not the first time that the conduct of members of the Secret Service had drawn intense scrutiny to the agency. In 2012, several agents resigned or faced their dismissal after revelations that they had engaged in misconduct while protecting President Barack Obama during an overseas trip to Colombia, including encounters with prostitutes.

The investigation of Mr. Taherzadeh and Mr. Ali began after a letter carrier with the United States Postal Service was assaulted in March at an apartment complex where the men had been living. A U.S. postal inspector went to the complex to interview witnesses, including the two men.

The men told the inspector that they were investigators with the U.S. Special Police Investigation Unit, according to the affidavit. They said they were part of an undercover investigation into gang-related activity as well as an inquiry into the Jan. 6, 2021, attack on the U.S. Capitol.

Other residents of the building told the inspector that the men had identified themselves as Homeland Security special agents, that they used other apartment units in the building and that they drove around in a black S.U.V. equipped with emergency lights that they described as an official Homeland Security vehicle.

The residents said that Mr. Taherzadeh and Mr. Ali had placed various pieces of surveillance equipment around the building and told residents that they had access to their cellphones and personal information, according to the affidavit. It was not clear why they told this to residents.

The inspector also learned that Mr. Taherzadeh and Mr. Ali were in regular contact with several members of the Secret Service who lived in the same building and that the men had given them gifts, Mr. Elias wrote.

The inspector reported the information to the Homeland Security Office of the Inspector General, which referred the case to the F.B.I.

One witness told investigators of being recruited by Mr. Taherzadeh and Mr. Ali to be a Homeland Security employee and serve on a task force. As part of the recruitment process, the men shot the witness with an Airsoft rifle “to evaluate their pain tolerance and reaction,” Mr. Elias wrote.

The men told the witness, who was not identified in the affidavit, that they needed to conduct research “on an individual that provided support to the Department of Defense and intelligence community,” according to the affidavit.

Mr. Taherzadeh often carried a Glock 19, Mr. Elias wrote.

He sent one Secret Service agent pictures of himself in tactical gear bearing the word “police.” In another photo, he posed in front of a stack of large cases that are often used to carry firearms. He also sent an agent a photo that he claimed showed a training session for Homeland Security Investigations.

“The investigation revealed that the photo Taherzadeh sent is a stock photo from the internet,” Mr. Elias wrote. There is no record that he had ever participated in such training, he the affidavit said.

Mr. Taherzadeh told a Secret Service member that he worked in a gang unit at Homeland Security and had credentials for U.S. Immigration and Customs Enforcement, according to the affidavit.

For about one year, he provided the employee a rent-free penthouse apartment that would have cost $40,200, the affidavit said.

“Although Mr. Taherzadeh was very outspoken about his job,” Mr. Elias wrote, “he claimed that he was part of a covert task force.”