Jeanine Pirro, the Fox News personality and longtime friend and champion of President Trump, has discussed with him the possibility of a book project to counter the damning portrayal of his presidency in Michael Wolff’s book “Fire and Fury,” according to three people with knowledge of their conversations.

The project has been described among West Wing advisers as “No Fire, No Fury,” according to two administration officials. Ms. Pirro met with Mr. Trump in the White House on Wednesday, hoping to secure his participation.

Over lunch, Mr. Trump agreed to be interviewed by Ms. Pirro for the book, according to one of the people briefed on their discussion.

For weeks, the president has seethed about the depiction of him in “Fire and Fury” as a volatile and ill-equipped chief executive, even asking the White House counsel and allies whether a libel lawsuit against Mr. Wolff is still possible. (His advisers have told him no.)