Citing an unnamed government official, the newspaper on Saturday said the warning came in a secret briefing that Bush received at his ranch in Crawford, Texas, on 6 August 2001.
A report by a joint Congressional committee last year alluded to a "closely held intelligence report" that month about the threat of an attack by al-Qaida, the paper said.
The disclosure appears to contradict the White House's repeated assertions that the briefing the president received about al-Qaida's threat was "historical" in nature and that the White House had little reason to suspect an al-Qaida attack within the US, The Times said.
The Congressional report said al-Qaida operatives had appeared to have a support structure in the US and intelligence officials had "uncorroborated information" that Bin Ladin "wanted to hijack airplanes" to gain the release of imprisoned "extremists".
It also said intelligence officials received information in May 2001, three months earlier, that indicated "a group of Bin Ladin supporters was planning attacks in the United States with explosives".
The paper said the White House had also offered evidence that the Federal Bureau of Investigation (FBI) had received instructions more than two months before the September 11 attacks to increase its scrutiny of suspects inside the US.
The disclosure appeared to signal an effort by the White House to distance itself from the FBI in the debate over whether the Bush administration did enough in the summer of 2001 to deter a possible attack in the US in the face of increased warnings, The Times said.
A classified memorandum sent around 4 July 2001, to Condoleezza Rice, the president's national security adviser, from the counterterrorism group run by Richard Clarke, described a series of steps it said the White House had taken to put the nation on heightened terrorist alert, according to the report.
The White House appears to
distance itself from the FBI
Among the steps, the memorandum said, "all 56 FBI field offices were also tasked in late June to go to increased surveillance and contact with informants related to known or suspected terrorists in the United States."
Parts of the White House memorandum were provided to The New York Times on Friday by a White House official seeking to showcase the administration's work against terrorism, the paper said.