WASHINGTON — As the F.B.I. began looking into allegations of sexual assault against Judge Brett M. Kavanaugh during his Supreme Court confirmation hearings last year, a Democratic senator wrote to the director of the bureau saying that he had “information relevant” to the inquiry, but that the F.B.I. apparently failed to follow up.

The letter, sent early last October by Senator Chris Coons, Democrat of Delaware, to Christopher A. Wray, the F.B.I. director, has come to light after a forthcoming book by two New York Times reporters surfaced a new allegation of sexual impropriety by a young Mr. Kavanaugh.

The revelations have reopened the bitter partisan debate over the confirmation of Justice Kavanaugh, just as he is coming up on his first anniversary on the Supreme Court, and raised fresh questions about how the F.B.I., under the direction of the White House, handled its investigation of the nominee.

Democrats, then and now, argue that the inquiry was insufficient and geared more toward clearing President Trump’s Supreme Court pick than toward uncovering the truth. Some of them, including several presidential candidates, seized on the new information to call for Justice Kavanaugh’s impeachment, or a new inquiry into the investigation that preceded his confirmation.