WASHINGTON — Exiled from Twitter, former President Donald J. Trump issued a sarcastic statement recently inquiring about the ongoing public silence from John H. Durham, the special counsel who has been investigating the Trump-Russia inquiry since May 2019.

“Where’s Durham?” said Mr. Trump, who repeatedly predicted before last year’s election that Mr. Durham’s investigation would prove a deep-state conspiracy against him. “Is he a living, breathing human being? Will there ever be a Durham report?”

Mr. Durham ignored the complaint publicly, and the scope of his inquiry remains opaque. But one aspect has come into focus recently, according to people familiar with the investigation: Mr. Durham has keyed in on the F.B.I.’s handling of a notorious dossier of political opposition research both before and after the bureau started using it to obtain court permission to wiretap a former Trump campaign adviser in 2016 and 2017 and questioned witnesses who may have insight into the matter.

In particular, Mr. Durham has obtained documents from the Brookings Institution related to Igor Danchenko, a Russia researcher who worked there a decade ago and later helped gather rumors about Mr. Trump and Russia for that research, known as the Steele dossier, according to people familiar with the request.

By asking about the dossier, Mr. Durham has come to focus at least in part on re-scrutinizing an aspect of the investigation that was already exposed as problematic by a 2019 Justice Department inspector general report and led to reforms by the F.B.I. and the Foreign Intelligence Surveillance Court.

A spokesman for Mr. Durham declined to comment.

Asked whether the special counsel had briefed his new supervisor — Attorney General Merrick B. Garland — a Justice Department spokesman would only point to a statement by Mr. Garland as a nominee. “If confirmed,” he said, “one of the first things I am going to do is speak with Mr. Durham and learn the status of his investigation.”

In February, several weeks before the Senate confirmed Mr. Garland, Mr. Durham obtained old personnel files and other documents related to Mr. Danchenko from the Brookings Institution, a prominent Washington think tank, using a subpoena. Mr. Danchenko had worked there from 2005 until 2010.

Mr. Danchenko traveled to Russia in 2016 and gathered rumors about Mr. Trump and his associates on behalf of Christopher Steele, who produced the dossier as a subcontractor for an investigative firm being indirectly paid by Democrats to look into any Trump-Russia ties.

Michael Cavadel, the general counsel of Brookings, confirmed the subpoena for records and other materials about Mr. Danchenko, saying that it was received on Dec. 31 and that the think tank had taken until February to gather the files and turn them over to Mr. Durham’s team in part because its office is closed during the pandemic.

“Consistent with its practices in such matters, Brookings provided the responsive documents, none of which contained information associated with the reports known as the Steele dossier,” Mr. Cavadel said.

Last September, the attorney general at the time, William P. Barr, made public that from 2009 to 2011 Mr. Danchenko had been the subject of an F.B.I. counterintelligence investigation assessing his contacts with several suspected Russian intelligence officials, including at the Russian Embassy.

(Skeptics of the Steele dossier have raised the prospect that Russian intelligence may have used Mr. Danchenko or his sources to seed it with disinformation, in order to further sow chaos. Mr. Danchenko was never charged and has denied ever being a Russian agent. He has also noted that during his time at Brookings he put forward analysis embarrassing to President Vladimir V. Putin of Russia: evidence that Mr. Putin plagiarized parts of his dissertation.)

Mr. Durham has also asked questions that suggested a focus on skepticism about how the F.B.I. approached issues that might have undermined the dossier’s credibility as a basis for wiretap applications, people familiar with the inquiry said.

For example, Mr. Durham’s team is said to have asked why the F.B.I., after identifying Mr. Danchenko as a major source for the dossier and interviewing him in early 2017, did not tell the surveillance court that he had once been the subject of a counterintelligence investigation.

Mr. Durham is also said to be interested in a meeting between the F.B.I. and Mr. Steele in Rome in early October 2016, shortly before the bureau submitted the first wiretap application that used information from his dossier.

The previous month, Yahoo News had published an article that contained information that overlapped with claims in the dossier, and the F.B.I. later learned that Mr. Steele had been a source for it, prompting the bureau to sever its relationship with him. At the time, as the bureau told the court in its wiretap application, it assumed the source had been someone else who had received a copy of the dossier.

Mr. Durham is said to have asked why F.B.I. officials at that October meeting apparently did not ask Mr. Steele whether he was the article’s source — before using his information to apply for permission to wiretap the former Trump adviser, Carter Page.

The focus raised the possibility that Mr. Durham has been exploring whether F.B.I. officials knowingly misled the surveillance court. But if Mr. Durham has found credible evidence of such a crime — as opposed to sloppy investigative work — he has yet to file any such charges.

Mr. Durham interviewed the former C.I.A. director John O. Brennan in August, but told him he was not the target of any criminal inquiry. But he has yet to interview former F.B.I. officials who held senior roles in 2016 and have been demonized by Trump supporters, including the former director James B. Comey; his former deputy Andrew G. McCabe; and a former senior counterintelligence agent, Peter Strzok, according to people familiar with the matter.

To the extent any eventual Durham report focuses on criticizing the F.B.I.’s handling of issues related to the Steele dossier, it would risk largely retreading ground already covered by the 2019 report by the Justice Department’s inspector general, Michael E. Horowitz.

Mr. Horowitz has already brought to light the fact that the F.B.I. botched its wiretap applications in numerous ways, including uncovering numerous material facts that law enforcement officials failed to tell the court and that might have undermined their case for receiving wiretap authorization or renewals — including about the dossier.

Mr. Horowitz’s report also already unearthed the fact that Mr. Danchenko had been the subject of a counterintelligence investigation when he worked at Brookings, in a footnote that was initially classified before Mr. Barr decided to make it public.

The report also already focused criticism on the F.B.I.’s failure to ask Mr. Steele in October 2016 whether he played a role in the Yahoo News article.

And to date, the misconduct by the only person Mr. Durham has charged — Kevin Clinesmith, a former F.B.I. lawyer who altered an email shown to a colleague during preparations to seek a renewal of the wiretap, preventing another problem from coming to light internally — was uncovered by Mr. Horowitz’s investigation. (Mr. Clinesmith, who pleaded guilty to falsifying the email but insisted that he did not deliberately mislead his colleague, was sentenced to probation.)

Mr. Barr assigned Mr. Durham to hunt for any potential wrongdoing by the Trump-Russia investigators in spring 2019, at a time when Mr. Trump and his supporters were pushing the notion that the inquiry had been a “deep state” plot against him. While Mr. Durham’s work has been opaque, accounts by people familiar with his investigation have made clear that he has pursued various Trumpian conspiracy theories and grievances.

In seeking to discredit the Russia investigation, Mr. Trump and his allies have frequently conflated it with the flawed Steele dossier. In fact, the Page wiretaps were a minor part of the overall effort, and Mr. Horowitz’s report showed that it played no role in the F.B.I. decision to open the counterintelligence investigation in July 2016.

While uncovering numerous ways the F.B.I. had botched those wiretap applications, Mr. Horowitz’s report also concluded that it had lawfully opened the overall investigation on an adequate basis. When the inspector general delivered the report, Mr. Durham intervened with an unusual public statement saying he disagreed with Mr. Horowitz that the investigation’s opening was properly predicated.

Mr. Durham provided no details, but Mr. Horowitz later told Congress that Mr. Durham had told him he thought that the F.B.I. should have opened the inquiry as a “preliminary” investigation rather than going straight to a “full” one.