According to an internal memo from Microsoft, the next Xbox will not need an internet connection to play games:
"Durango [the codename for the next Xbox] is designed to deliver the future of entertainment while engineered to be tolerant of today's Internet. There are a number of scenarios that our users expect to work without an Internet connection, and those should 'just work' regardless of their current connection status. Those include, but are not limited to: playing a Blu-ray disc, watching live TV, and yes playing a single player game."
This puts the final nail in the coffin for that naughty little rumor about the next Xbox constantly requiring an internet connection. That being said, you might need to be online to install your games. If that’s the case, the setup will be incredibly similar to Steam, where you need to be online to download and install your games before having the ability to play them offline.
Source: Ars Technica