The White House’s review of whether to release a transcript of a special counsel’s interview of President Biden that set off a political furor is being complicated by the sensitive material it covers, including classified information, security measures and discussions that could be subject to executive privilege, people familiar with the matter said.

The White House has been pressed by reporters seeking the transcript since the release last week of the report by Robert K. Hur, a special counsel who investigated Mr. Biden’s handling of classified records from his vice presidency after he left office. And three Republican chairmen of House oversight committees have sent a letter to Attorney General Merrick B. Garland demanding that he turn over both the transcript and audio recording.

But while the White House has indicated that it is looking at releasing the material to support its assertions that Mr. Hur’s characterization of Mr. Biden’s memory as having “significant limitations” was inaccurate and driven by partisanship, it has made no commitment to do so and has offered no timetable. It does not appear that such a disclosure is imminent, according to people who spoke on the condition of anonymity to discuss sensitive internal matters about why the review is challenging.

Mr. Hur, who was appointed by Mr. Garland and had previously served as a political appointee in the Trump Justice Department, found that “no criminal charges are warranted” against Mr. Biden even though classified material from his vice presidency had been found at an office and in his home.

While the report said there was some evidence consistent with a conclusion that Mr. Biden had willfully retained the files without authorization, it also said the facts fell short of proving that he did so and other evidence was consistent with innocent explanations.

But even as Mr. Hur concluded that there was no criminal case to be brought, he also portrayed Mr. Biden as an “elderly man with a poor memory” who in their interview forgot what years his term as vice president began and ended and what year his son Beau died.

In a letter appended to the report, Mr. Biden’s government and personal lawyers — who were present for the interview — sharply disputed Mr. Hur’s portrayal of the president’s memory as unusually faulty, calling it both gratuitous and inaccurate. The pushback immediately raised the question of what the full transcript actually showed.

On Feb. 9, the day after the report was made public, Ian Sams, a spokesman for the Office of the White House Counsel, told reporters that it was a “reasonable question” to ask whether the transcript would be released and said they would “have to work through” issues like the fact that the interview dealt with classified materials. Since then, the White House has pointed back to his remarks and declined to comment further.

But people familiar with the matter said classified material was just one of several complexities raised by the review, each of which is associated with its own potential bureaucratic process.

On the matter of classified information, the transcript was deemed highly classified from the start because the interview touched on systems for how classified material is stored and used by presidents, vice presidents and their staffs. Some of the topics discussed in the documents found in Mr. Biden’s office and home were also sensitive.

To undergo a declassification review, the transcript would need to be sent to various security-related agencies that have a stake in topics that were discussed for their review and input on whether anything would need to be redacted before any public release, the people noted. Those agencies potentially include the Pentagon, the State Department, the Office of the Director of National Intelligence and the C.I.A.

Another problem is that a major focus of the investigation was Mr. Biden’s home in Delaware. To the extent the interview went into matters like the layout of rooms in his house and its security measures, those portions of the transcript apparently need to be reviewed by law enforcement and Secret Service officials to ensure that no material is made public that could jeopardize the security of the president and his family.

Yet another issue, the people said, involves executive privilege. While Mr. Biden, advised by his lawyers, declined to assert the privilege to block the release of anything Mr. Hur chose to include in his report, the five-hour interview was much wider ranging than the selections the special counsel cited.

Mr. Hur apparently asked Mr. Biden about deliberations within the Obama administration about foreign policy matters. A major focus of the report was a memo Mr. Biden had sent to President Barack Obama opposing the latter’s decision in 2009 to send a surge of additional troops into the Afghanistan war zone, for example.

To the extent that the interview included discussion of Mr. Biden’s conversations with Mr. Obama, vice president to president, or broader deliberations within the cabinet and the White House foreign policy team during the Obama administration, the people said, the transcript could contain matters that Mr. Biden would choose to keep secret as privileged.

The doctrine of executive privilege allows the executive branch to keep some material about high-level internal deliberations secret from Congress or the courts to avoid chilling the candor of future internal discussions about how presidents should carry out their duties.

But especially since members of Congress have asked for the transcript, any such redactions could later lead to litigation. For example, the House could subpoena the unredacted version of the transcript and then sue if Mr. Garland, citing any assertion of executive privilege by Mr. Biden, declined to turn it over.

Against that backdrop, the process for formally asserting executive privilege is multifaceted. Among other things, typically such material is submitted by the White House to the Justice Department’s Office of Legal Counsel for a review and its analysis of whether particular things are legitimately covered by the privilege as a legal matter.