The New York Times said Bolton urged Annan to "personally and publicly" reject the remarks by Mark Malloch Brown, the deputy secretary-general, chastising the United States for withholding funds from the world body.
Bolton called the remarks "very, very grave."
"The prevailing practice of seeking to use the U.N. almost by stealth as a diplomatic tool while failing to stand up for it against its domestic critics is simply not sustainable," Malloch Brown said in a speech Tuesday. "You will lose the U.N. one way or another."
Through a spokesman, however, Annan declined Bolton's demand.
"The secretary general stands by the deputy secretary general and agrees with the thrust of the speech. This is not a criticism of the United States, it is call for greater U.S. involvement in the U.N.," the spokesman said.
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