NBC is expected to part ways with “Today” show anchor Billy Bush after the emergence of a tape in which he engaged in a lewd conversation about women with Republican presidential candidate Donald Trump, a person familiar with the situation said.

The network is negotiating Mr. Bush’s exit, which could be finalized in the next several days, the person said. He had already been suspended indefinitely from his job co-anchoring the 9 a.m. to 10 a.m. hour of “Today.”

An NBC spokeswoman declined to comment.

The tape, which became public late last week, was recorded in 2005 when Mr. Bush was co-host of NBC’s entertainment show “Access Hollywood.” The controversial conversation happened while he was accompanying Mr. Trump to the set of a soap opera where the real estate mogul was making a cameo.

On the tape, Mr. Bush egged on Mr. Trump while the businessman boasted of his prowess with women, and laughed along when Mr. Trump described his failed efforts to have sex with Nancy O’Dell, who was Mr. Bush’s co-host of “Access Hollywood” at the time.

Ms. O’Dell has since condemned the remarks, saying in a statement that “there is no room for objectification of women or anybody for that matter, not even in the `locker room.’”

Mr. Trump has apologized for the comments, while categorizing them as “locker-room banter.”

In an email to staff on Sunday, “Today” executive producer Noah Oppenheim said, “There is simply no excuse for Billy’s language and behavior on that tape.”

Mr. Bush has apologized for the remarks, saying he was “embarrassed and ashamed.”

Given that the primary audience for “Today” is women, the network may have been concerned about Mr. Bush’s damaged reputation being a drag on viewership. His Facebook page has been filled up with critical comments from users, many of whom have called for his removal from the show.

Even some Hollywood publicists, who “Today” counts on having close relationships with for access to stars, have come out publicly against Mr. Bush on social media.

Mr. Bush, 44, has been a rising star at NBC and was recently named co-anchor of “Today.” He is a cousin of former president George W. Bush.

NBC also came under scrutiny for its handling of the “Access Hollywood” remarks as a news story. NBC News and “Access Hollywood” have said they were working on their own stories but were slowed down by internal legal reviews. The story was leaked to the Washington Post, which first published the lewd video.

Write to Joe Flint at joe.flint@wsj.com