Bush, who was a host with Access Hollywood at the time of the recording, will be suspended "indefinitely," the Associated Press reported Sunday.
"There is simply no excuse for Billy's language and behavior on that tape," Today executive producer Noah Oppenheim said in an internal memo.
Following the tape's leak, Bush released a statement saying he was "embarrassed and ashamed."
"I was younger, less mature and acted foolishly in playing along. I'm very sorry," Bush said in the statement. Bush was 32 years old when the tape was recorded.
In the recording, Bush can be heard laughing along to Trump's comments, saying "you can do anything [to women]" when you're a celebrity.