Bush's involvement has turned into a major problem as morning-show audiences are not always forgiving.
The development was captured on a show memo seen by Reuters on Sunday. It is unclear for exactly how long Bush will be suspended.
On the debate stage, Clinton did not respond directly to Trump's accusations about her husband or her own role, but was blistering in her condemnation of his predatory comments about women in the tape released Friday. Bush can be heard on the tape laughing and joking with Trump.
Trump entered Sunday night's debate desperate to steady his floundering campaign.
Although there has been no official word on whether Bush will be back on the show, which he joined recently as a regular co-host, there is speculation lawyers at NBC are working on an exit agreement for him.
Bush was brought aboard Today to lend a different tone to the 9am hour of the show, which competes directly in many markets with the popular syndicated program Live, featuring Kelly Ripa. "Since then, we have sunk so low we are breaking through the Earth's crust, where drowning in boiling magma will come as sweet sweet relief", Oliver said.
In an internal memo to NBC staff on Monday, senior vice president Noah Oppenheim - who oversees the Today show - said there was "no excuse for Billy's language".
Republican presidential nominee Donald Trump and Democratic presidential nominee Hillary Clinton during the second presidential debate at Washington University on Sunday in St. Louis, Missouri. Bush was expected to address the lewd comments he made while chatting with Donald Trump in a 2005 leaked video as they prepped for an Access Hollywood segment. One of those women, "Days of Our Lives" actress Arianne Zucker, broke her silence on Sunday.
Angering women viewers of a show as important to NBC's bottom line as "Today" is no small matter.
As you are probably aware, [he was] part of that 2005 Access Hollywood taping with Donald Trump that's been in the news all weekend.
Bush issued a forthright apology statement on Friday, but it did not quiet the calls for action by NBC.
The thing about the entire situation is that Billy Bush claimed that he was young and foolish at the time of the recording, despite having been in his thirties at the time.
Bush called the actress "hot", adding an expletive.
"Obviously I'm embarrassed and ashamed", he explained. Anyone who knows me knows these words dont reflect who I am, Trump said.
"I think there's kind of a piece of the jigsaw puzzle missing in Donald Trump where he does not look at women and consider them as equal to himself", Kaine said on CNN. It's like a magnet. Bill Clinton has said far worse to me on the golf course - not even close. And when you're a star, they let you do it. "You can do anything". "Grab 'em by the p-y".
The Trump pre-debate event was the clearest sign yet that he planned to use the former president's sexual history to try to distract from the swirling controversy over his own predatory remarks about women.