As the hunt for Osama bin Laden continues on both sides of the border between Afghanistan's Tora Bora district and the adjacent tribal regions of Pakistan, a succession of speculative and unsubstantiated reports have surfaced suggesting that the Qaeda leader may already be dead as a result of American bombing or even illness.

Over the last three days, the suggestion has come from Pakistan's president, Gen. Pervez Musharraf, from Gen. Tommy R. Franks, the American military commander for Afghanistan, and from Kenton Keith, the spokesman here for the antiterror campaign -- as well as in today's issue of a Pakistani newspaper, The Pakistan Observer.

In the case of President Musharraf, General Franks and Mr. Keith, the statements were conjecture, based on the intensity of the bombings at Tora Bora, not on any tangible evidence of Mr. bin Laden's death.

Only The Pakistan Observer went further, with a front-page report under an Islamabad dateline that quoted an unnamed Taliban leader as saying that Mr. bin Laden ''had a peaceful natural death in mid-December in the vicinity'' of the Tora Bora mountains. The report said that his death was the result of a ''serious lung complication.''