President Barack Obama rebuffed Geithner's suggestion, and he remained at Treasury until 2013.
Geithner's memoir will be published next week. Yesterday, The Associated Press bought an early copy.
Geithner writes of the incident in "Stress Test," which explores his turbulent four years at Treasury. During his tenure, the Obama administration faced the worst recession and most severe financial crisis since the Great Depression.
In several sections of his memoir, Geithner recounts internal conversations about who might replace him in a post he was initially reluctant to accept.
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In proposing that the White House consider Clinton as his successor later in 2010, Geithner cited her star power as secretary of state. Among the other names Geithner suggested was Jack Lew, who succeeded him last year.
No Treasury secretary since the Depression confronted so many financial threats at once. When Geithner became Treasury secretary in January 2009, the economy had sunk into a deep recession. Unemployment was surging, and the financial system was teetering.
Geithner's supporters said he deserved credit for helping steady the banking system, restore investor confidence and avert a complete meltdown.
His critics countered that Geithner's policies consistently favoured big banks and neglected ordinary Americans, including many struggling to save their homes after a wave of foreclosures followed the housing bust.
He acknowledges that the government's efforts to help troubled homeowners keep their homes were inadequate. But he argues that no "game-changing housing plan" that would have offered Americans substantial relief existed for Geithner's team to embrace.