"Of course, the president and his family had secret service protection. My family did not. And it was a somewhat awkward moment in the meeting," he added in describing a 2003 national security meeting when he was told of the plot.
Rumsfeld, who has two daughters and a son, said Bush urged him to take the threat seriously because US forces had killed Saddam's sons Uday and Qusay and his 14-year-old grandson Mustapha in July 2003.
"I made a comment like 'Thank you' or something and president Bush looked me in the eye and said 'You better take this seriously,'" the former Pentagon chief recalled.
"And of course, I did take it seriously. But I was also realistic that there was not much one could do about that."
The television appearance was part of Rumsfeld's effort to promote his new memoir, "Known and Unknown," which was released earlier and recounts his long government career serving Republican presidents from Richard Nixon to Bush.
He has also released online a wide range of nearly 2,000 documents from his tenure in public service, dating back to his years as an Illinois congressman in the 1960s.
Many of them paint him in a positive light -- showing concern for military interrogations, for example -- that stand in sharp contrast to his public statements at the time.
Rumsfeld has also joined massively popular micro-blogging website Twitter and social network service Facebook.
His Twitter handle, @RumsfeldOffice, tweeted on Sunday that "100 years after president (Ronald) #Reagan's birth, it is worth remembering that the Cold War didn't just end. It was won."
He has nearly 6,000 followers since joining the website in October.
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