Ed Martin, the ardent Trump loyalist serving as interim U.S. attorney in Washington, is pursuing an inquiry into whether former President Joseph R. Biden Jr. was competent to pardon his family members and others during his final days in office.

The inquiry, which includes previously unreported letters to Mr. Biden’s family and former White House staff members, uses the levers of federal law enforcement to try to harass Mr. Biden’s family and allies and undermine his decisions as president, while scoring political points with President Trump.

Mr. Trump and his supporters have increasingly seized on the unsubstantiated theory that the pardons Mr. Biden issued during his final months in office may be invalid because he lacked the mental capacity to consent to them.

In an apparent effort to build the case, Mr. Martin began sending letters two months ago to Mr. Biden’s White House aides, including Jeffrey D. Zients, the former chief of staff, as well as recipients of Mr. Biden’s pardons, such as his brother James Biden and sister-in-law Sara Biden, according to three people familiar with the effort who were not authorized to discuss it publicly, and a copy of a letter obtained by The New York Times.

The letters, signed by Mr. Martin, are informal but provocative, questioning a presidential clemency power that has generally gone unchallenged. They highlight Mr. Martin’s hands-on approach and willingness to use one of the most important U.S. attorneys’ offices in the country to seek retribution against Mr. Trump’s perceived enemies.