On Monday, executive Noah Oppenheim wrote in a memo to the morning show's staff that there is "simply no excuse for Billy's language and behaviour on that tape".
Mr. Bush has said he was "embarrassed and ashamed" at the remarks.
In the video, Mr Trump was on a bus talking to Mr Bush - then host of NBC's Access Hollywood - ahead of a segment they were about to record.
Mr Trump says "you can do anything" to women "when you're a star" and also brags about trying to grope and kiss women.
Mr Bush tells Mr Trump at one point "the Donald has scored" and also comments on a woman's "good legs".
The presenter, who is the first cousin of former President George W Bush, has since said: "It's no excuse, but this happened 11 years ago - I was younger, less mature, and acted foolishly in playing along. I'm very sorry."
Top Republicans have condemned Mr Trump for the comments. His election rival Hillary Clinton called them "horrific".
Bush joined the Today team in August, having been anchor of Access Hollywood since 2009 and co-host of Access Hollywood Live since 2010.
Mr. Oppenheim said that there would be a "further review" of the matter and did not say for how long Mr. Bush would be suspended.
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