Mr. Mueller has also been examining a false statement that the president dictated on Air Force One in July in response to an article in The Times about a meeting that Trump campaign officials had with Russians in 2016. A new book, “Fire and Fury: Inside the Trump White House,” by Michael Wolff, says that the president’s lawyers believed that the statement was “an explicit attempt to throw sand into the investigation’s gears,” and that it led one of Mr. Trump’s spokesmen to quit because he believed it was obstruction of justice.