The announcement on Tuesday came after a week rife with speculation over the likelihood of a Bush cabinet reshuffle.
Citing government officials on Thursday, The New York Times had predicted Ashcroft, 62, and Homeland Security Secretary Tom Ridge, 59, would be among those likely to step down for personal reasons.
Other members of the Bush cabinet also expected to leave their posts in the next four years are Secretary of State Colin Powell, 67, and Defence Secretary Donald Rumsfeld, 72.
Bush claimed election victory on 3 November after Democratic challenger John Kerry conceded the first presidential race since the 11 September 2001 attacks on the US.
The Massachusetts senator had held out for hours in the hope he could pull out a miracle win in the pivotal state of Ohio.