WASHINGTON — The Justice Department will allow John H. Durham to remain in the role of special counsel appointed to investigate the origins of the Trump-Russia inquiry, even after he relinquishes his role as the top federal prosecutor in Connecticut, according to a senior Justice Department official.

Mr. Durham is expected to step down as the U.S. attorney in Connecticut as early as Tuesday, when the Biden administration will begin to ask dozens of Trump-era U.S. attorneys who have not already quit to submit their resignations, the official said Monday.

All of the remaining U.S. attorneys appointed by President Donald J. Trump and confirmed by the Senate will be asked to tender their resignations except for David C. Weiss, the U.S. attorney in Delaware who is overseeing the tax fraud investigation into President Biden’s son Hunter Biden. Acting Attorney General Monty Wilkinson called Mr. Weiss on Monday evening and asked him to remain in office, according to the official.

It is common for new presidents to replace U.S. attorneys en masse, and the request for resignations has long been expected. But Mr. Durham’s and Mr. Weiss’s investigations had created delicate situations for the Biden administration, which is seeking to restore the Justice Department’s image of impartiality.

It is not clear exactly when the resignations, 56 in all, will take effect, or when their replacements can be confirmed by the Senate. The resignations were reported earlier by CNN.

The confirmation hearing for Judge Merrick B. Garland, Mr. Biden’s nominee for attorney general, is not expected to begin for two weeks, according to a person briefed on the matter. The process has been slowed by the tumultuous transition from the Trump administration and by the second impeachment trial of Mr. Trump, which begins on Tuesday.

Since the spring of 2019, Mr. Durham has been investigating whether any Obama administration officials broke the law when they examined the Trump campaign’s potential ties to Russian efforts to interfere in the 2016 presidential election.

Both Mr. Trump and the attorney general at the time, William P. Barr, had publicly said they were certain that Mr. Durham would uncover grave offenses, if not outright criminal behavior, that supported the idea that the Russia investigation was a plot created to sabotage Mr. Trump.

But Mr. Durham never lived up to their expectations. The only criminal case Mr. Durham has brought was against Kevin E. Clinesmith, a former lower-level F.B.I. lawyer, who falsified information in an email from the C.I.A. that the bureau used to renew a wiretap order that targeted Carter Page, a onetime Trump campaign aide. In the weeks before the 2020 election, Mr. Trump and his supporters expressed outrage that the Durham inquiry had not produced anything useful to Mr. Trump’s campaign efforts.

In October, Mr. Barr secretly appointed Mr. Durham to serve as special counsel to continue his work. The move gave Mr. Durham independence from a possible Biden administration and made it very difficult for a new attorney general to end his investigation, all but ensuring the Durham inquiry would live on after Mr. Trump left office.

“In advance of the presidential election, I decided to appoint Mr. Durham as a special counsel to provide him and his team with the assurance that they could complete their work, without regard to the outcome of the election,” Mr. Barr wrote in a letter that he submitted to Congress in December.

Dozens of Mr. Trump’s U.S. attorneys have already resigned, in the weeks before and after the election, leaving those offices in the hands of acting officials. While Mr. Durham and several more U.S. attorneys are expected to join them this week, that cohort will not include the leaders of the largest, most prominent federal prosecutor’s offices: Audrey Strauss, the U.S. attorney in Manhattan, who was appointed to her position by the courts, and Michael R. Sherwin, the U.S. attorney in Washington, who is an acting official and was not confirmed by the Senate.

Both Ms. Strauss and Mr. Sherwin were elevated to their roles amid upheaval and controversy that stemmed from Mr. Barr’s handling of politically delicate cases involving Mr. Trump.

Ms. Strauss was made the acting U.S. attorney after her boss, Geoffrey S. Berman, angered the White House with his handling of cases against Mr. Trump’s associates and ultimately refused to leave when Mr. Barr tried to replace him. The standoff between the two men ended when Mr. Barr allowed Ms. Strauss, a registered Democrat, to lead the office. Federal judges in her district, exercising a rarely used power, formally appointed her to the position in December.

Mr. Sherwin was tapped to lead the Washington office after his predecessor was removed amid a contentious decision by Mr. Barr to force prosecutors to lower a sentencing recommendation for one of Mr. Trump’s allies, Roger J. Stone Jr. Mr. Sherwin has since emerged as the face of the Justice Department’s sprawling investigation into the Jan. 6 siege of the Capitol by a pro-Trump mob.

Mr. Sherwin could remain at the department to work on the Capitol riots investigation, even after the administration nominates a new U.S. attorney, according to a person with knowledge of the deliberations.