On one of his last campaign stops before Tuesday's presidential election, Mr Bush accused his rival of choosing "the path of weakness and inaction".
For his part, Mr Kerry said &to=http:// english.pravda.ru/usa/2002/12/21/41144.html ' target=_blank>President Bush had missed the chance to capture Bin Laden when he was cornered.
The exchanges came a day after &to=http:// english.pravda.ru/main/2002/09/13/36520.html ' target=_blank>Bin Laden warned of more attacks on the US.
In a video message broadcast on al-Jazeera Arabic television, he accused Mr Bush of deceiving the &to=http:// english.pravda.ru/letters/2002/02/23/26619.html ' target=_blank>American people in the three years since the 11 September 2001 attacks.
"The reasons are still there to repeat what happened," he said, BBC News reported.
According to Reuters, Kerry, who dropped the reference to Tora Bora from his stump speech after the tape surfaced on Friday night, also went back to his familiar pitch on Saturday. He criticized Bush for diverting the military's focus from pursuing bin Laden to invading Iraq.