Mr. Trump, who values the control of information and seeks to shape his image at all times, had aides on the campaign sign nondisclosure agreements, the same type of measure he has used for employees at his private real-estate company for decades. The first section of the nondisclosure agreement says no confidential information is to be released during the term of service or anytime “thereafter.”

Ms. Manigault Newman, who was the communications director for the Office of Public Liaison in the White House until being fired in December, has released tapes over several days that revealed her private conversations with Mr. Trump and other officials connected to him. That has stoked interest in the book and also called into question denials from Mr. Trump’s team. Most of her news media appearances have been on NBC, the network that vaulted her to prominence on “The Apprentice” with Mr. Trump more than a decade ago.

On Monday night, Katrina Pierson, a spokeswoman for the Trump campaign, denied on Fox News that the president had used the N-word, as Ms. Manigault Newman claims in the book. But on Tuesday, Ms. Manigault Newman provided an audio recording to CBS that appeared to reveal Ms. Pierson saying during the campaign that she believed Mr. Trump had used the slur, and that he was embarrassed for having done so.

At the White House briefing on Tuesday, the press secretary, Sarah Huckabee Sanders, said she could not guarantee that a tape would never emerge with Mr. Trump using the slur. Ms. Manigault Newman claims such a tape exists from their days on “The Apprentice.”

Until excerpts from the book began leaking out on Friday, the White House had gotten Mr. Trump to agree to a plan to ignore it, in the hopes of depriving it of oxygen. That plan called for the opposite of what Mr. Trump did when Michael Wolff published “Fire and Fury,” a story that had mostly been told in real-time news media accounts, but which the president stoked interest in by tweeting about it and trying to block its release.