Rudolph W. Giuliani, the former mayor of New York, seemed in a giddy mood when he was interviewed last week on the “Fox & Friends” morning television show.

Tireless if often wildly inaccurate in his attacks on Hillary Clinton’s ethics, health and work as a United States senator and as secretary of state, Mr. Giuliani has been spending every minute in the public spotlight as a surrogate for Donald J. Trump.

His most remarkable claim is that he has a pipeline into the Federal Bureau of Investigation and that agents tell him they are “outraged” that they have not been able to bring Mrs. Clinton and her husband, former President Bill Clinton, to justice.

But his television appearance on Tuesday of last week appeared, at the time, to be in a softer hue.

Brian Kilmeade, a Fox News host, asked Mr. Giuliani about the presidential campaign during its last two weeks.