NBC is in damage control regarding its handling of an 11-year-old recording of on-air personality Billy Bush and Republican presidential candidate Donald Trump talking in crude and misogynistic terms about women.

Mr. Bush’s exit is being negotiated as a result of an NBC News review into the matter that determined he should have been more forthcoming about the existence of the recording, according to executives familiar with the situation. There are still unanswered questions regarding how long “Access Hollywood” may have been...