According to a tweet posted via the Xbox Support account, Microsoft is working on adapter to make Xbox 360 headsets compatible with the upcoming Xbox One.
When asked if existing Xbox 360 headsets will work with the Xbox One on Twitter, the Xbox Support account replied that Microsoft are working on an adapter.
@craigfellows We are working to develop an adapter for current wired headsets to connect. More details as we approach launch. ^PS— Xbox Support (1-5) (@XboxSupport) June 26, 2013
While the Xbox One will not come with a headset, Microsoft has said that the Kinect bundled with every Xbox One will act as a stand in.