CBS News correspondent Peter Maer reports that a senior official indicates the White House hopes to release the memo later Friday.
PDBs are a compilation of information from law enforcement and intelligence agencies to keep the president updated on threats around the world. They normally are circulated among only a select group of top administration officials.
Commissioners already have seen the Aug. 6, 2001, PDB about bin Laden but want it released to the public "because we feel it's important that the American people get a chance to see it," commission chairman Thomas Kean said.
Meanwhile, former President Clinton defended his counterterrorism policies in a private meeting with the Sept. 11 commission and said intelligence wasn't strong enough to justify a retaliation against al Qaeda for the 2000 bombing of a Navy ship.
Mr. Clinton met for nearly four hours with the 10-member bipartisan panel in a closed-door session shortly after the conclusion of national, broadcast live on national television.
Commissioners described Mr. Clinton's testimony as frank and informative.
A person familiar with the session said Mr. Clinton told the commission he did not order retaliatory military strikes after the bombing of the USS Cole in October 2000 because he could not get "a clear, firm judgment of responsibility" from U.S. intelligence before he left office the following January.
It wasn't until after the Bush administration took power that U.S. intelligence concluded al Qaeda had sponsored the attack on the ship in the harbor at Aden, Yemen. Some commissioners have been critical of the decision not to launch a retaliatory military strike.
The person, who would speak only on condition of anonymity because Mr. Clinton's testimony delved into classified materials, also said the former president explained the rationale for many of the terror-fighting policies that his administration instituted and the message his administration left behind to the incoming Bush administration.
Mr. Clinton "did not indicate anything fundamentally that he would have done differently" given what U.S. intelligence knew about Osama bin Laden and the al Qaeda threat, the person said.
The panel said it didn't plan to release details of the meeting, saying much of it involved classified information.
"He did very well," commissioner John Lehman, a former Navy secretary under President Reagan, told CNN. "He gave us a lot of very helpful insights into things that happened, policy approaches."
Former Vice President Al Gore is scheduled to meet the panel soon.
Mr. Bush and Vice President Dick Cheney also will meet privately with the full panel in a joint session in coming weeks. They initially restricted the interview to one hour with two panel members, but under mounting public pressure agreed last week to a joint session without time constraints.
In her much anticipated, nationally televised testimony Thursday, Rice emphatically assigned blame for the pre-Sept. 11 failures on "frustratingly vague" U.S. intelligence, setting the stage for the top men at the CIA and FBI to explain next week what went wrong and what's been done to fix it.
"I really don't believe that all of our work is done, despite the tremendous progress that we've made thus far," Rice testified.
Next week, the bipartisan panel will examine law enforcement and intelligence failures surrounding Sept. 11, with scheduled testimony from Attorney General John Ashcroft, CIA Director George Tenet and FBI Director Robert Mueller, as well as from former FBI Director Louis Freeh and former Acting FBI Director Thomas Pickard.
"This hearing will focus on four important questions," Kean said. "How was our government structured before 9-11 to address the terrorist threat inside the United States? What was the threat in 2001 and our government's response to it? How did the intelligence community address the threat? What reforms have been taken since 9-11 to respond to the terrorist threat inside the United States?"
Rice testified in the wake of an appearance last month by former counterterrorism director Richard Clarke in which he said nothing would have prevented Sept. 11, but faulted the Bush administration for failing to take the al Qaeda threat seriously.
During Thursday's hearings, Rice said Mr. Bush "understood the threat" posed by bin Laden's group. His administration's first new national security policy, agreed days before the Sept. 11 attacks, was a plan to destroy al Qaeda, Rice said.
But there was no "silver bullet" that would have stopped 19 hijackers from seizing four planes and piloting them into the World Trade Center, the Pentagon and a field in Pennsylvania, Rice argued. She blamed decades of U.S. policy.
"America's response across several administrations of both parties was insufficient," Rice said, adding, "tragically, for all the language of war spoken before September 11th, this country simply was not on a war footing."
But Democratic commissioners sharply questioned Rice over signals that the administration may have received of the coming attacks.
They asked why alarms didn't ring when Bush was presented with the Aug. 6 classified memo, entitled "Bin Laden determined to attack inside United States."
Former Sen. Bob Kerrey, a Democratic member of the commission, described the memo as saying that "the FBI indicates patterns of suspicious activity in the United States consistent with preparations for hijacking."
Rice dismissed the document as "historical information based on old reporting" and said it did not warn of attacks inside the United States.
A CBS News poll found Rice's testimony improved her public image, but left most Americans — three in four —