Mr Schmidt's request was rejected as unacceptable by Sheryl Sandberg, who served as Google's vice-president of global online sales and operations for six years before leaving in March 2008 for Facebook, the book reportedly says.
Google announced in January that Mr Schmidt would be replaced as chief executive on April 4 by Google co-founder Larry Page.
Mr Schmidt, who openly endorsed Democratic candidate Barack Obama during the 2008 presidential election, will remain with Google as executive chairman.
The Times reported the book also details Google's troubled relationship with China, saying it was plagued by "missteps from the start".
Google announced in January last year that it had been targeted by cyber attacks originating in China and that it was no longer willing to self-censor content to comply with government rules.
In 2004, Google co-founders Mr Page and Sergey Brin were coached on how to behave during a visit to China, including receiving advice from former US vice-president Al Gore, the book says.
After formally entering China in 2006, Google fired its head of government relations there for giving iPods to Chinese officials and charging them to her Google expense account, the book says.
Google also reportedly refused to grant money to advertise in China and Mr Page and Mr Brin did not visit the country after Google opened an office there.
In addition, Google blocked software engineers in China from having access to its code base used to invent new products because it feared government officials might force them to reveal private information, the book says.
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