Former White House counter-terrorism tsar Richard Clarke said the Bush administration sanctioned the repatriation of about 140 high-ranking Saudi Arabians, including relatives of the al-Qaida chief.
"Somebody brought to us for approval the decision to let an aeroplane filled with Saudis, including members of the Bin Laden family, leave the country," he said.
Mr Clarke said he checked with FBI officials, who gave the go ahead. "So I said: ‘Fine, let it happen.’"
He first asked the bureau to check that no-one "inappropriate" was leaving.
"I have no idea if they did a good job," he added.
Dale Watson, the FBI’s former head of counter-terrorism, said that, while the bureau identified the Saudis who were on the plane, "they were not subject to serious interrogations".
The plane is believed to have landed in ten US cities picking up passengers, including Los Angeles, Washington DC, Boston and Houston. At the time, access to US airspace was restricted and required special government approval.
Tom Kinton, director of aviation at Boston’s Logan Airport, said: "We were in the midst of the worst terrorist act in history and here we were seeing an evacuation of the Bin Ladens."
But he said it was clear the flight had been sanctioned by federal authorities.
Prince Bandar bin Sultan, the Saudi ambassador to the US who is said to have organised the exodus, met President George Bush on September 13, 2001, two days after the terror attacks. It is not known if they discussed the repatriation plan.
The White House has declined to comment on the claims, but sources said the Bush administration was confident no secret flights took place.
Mr Clarke said he did not recall who requested approval for the flights, but believes it was either the FBI or the State Department.
But FBI spokesman John Iannarelli said: "I can say unequivocally that the FBI had no role in facilitating these flights."