For Billy Bush, the presidential cousin and genial Hollywood interviewer, 2016 was supposed to be a transformative year.

But not like this.

Just weeks after starting a prestigious gig as a host on NBC’s “Today” show, Mr. Bush on Friday found himself cast in a harsh and unflattering light — and facing scattered calls online to resign — after a recording emerged of him exchanging lewd and misogynistic remarks about women with Donald J. Trump.

In the recording, from 2005, Mr. Bush can be heard laughing as Mr. Trump describes in vulgar terms his efforts to seduce a married woman. The men, who were taping a segment for “Access Hollywood,” then boisterously admire the looks of a nearby television actress, with Mr. Bush shouting, “Yes, the Donald has scored!”

When Mr. Trump boasts that as a celebrity, he “can do anything” with women, another voice, apparently Mr. Bush’s, laughs and says, “Whatever you want.”