People who think they know Billy Bush may be surprised this week when he unveils a new video podcast.

“What this show will deliver is the real me — real unadulterated, unfiltered opinion, no fear, a great sense of humor,” says Bush during an interview. “I can’t wait, and I don’t give a s—t. I’m the head of H.R.”

Bush is known for his many years of hosting entertainment-news programs such as NBCUniversal’s “Access Hollywood” and, most recently, Warner Bros. Discovery’s “Extra.” But he sees an opportunity to put on an unbridled show that will...