Could the Xbox 360's successor feature a Radeon-derived graphics chip? Quite possibly, judging from a Fudzilla report that quotes "industry sources" as saying Microsoft has picked AMD to develop its next-gen console GPU.
The existing Xbox 360 already has AMD (or, rather, ATI) graphics inside: Xenos, a custom GPU design with Shader Model 3.0 support and an auxiliary chip that speeds up antialiasing. According to an old Beyond3D article, Microsoft orders Xenos chips from foundries directly, so the original deal may have involved an intellectual property license.
Fudzilla doesn't say much about the type of GPU Microsoft will include in its next console, although it does note that the third-gen Xbox won't arrive until 2012. Based on that schedule, the site speculates that the future graphics chip could be based on 28-nm process technology.
Aside from Microsoft, the pre-merger ATI also partnered up with Nintendo to provide the graphics technology inside the Wii. However, the Wii's Hollywood GPU is quite a bit slower and presumably based on older technology than Xenos, as well.