The program, called "Linuxpreview," causes the GameCube to draw an on-screen picture of Tux, the penguin Linux logo.
The leaders of the group, some of whom come from a project to port Linux to Microsoft's Xbox, were not immediately available to comment.
On their home page, project leaders suggested a Linux-powered GameCube could eventually be used as a server, a multimedia terminal or a desktop client computer connected to a server.
The console is powered by a version of IBM's PowerPC chip and is considered the least powerful of the three major game consoles, although the Web site notes "as it is a computer with decent RAM and a good CPU, it makes sense to port Linux to this platform."
A spokeswoman for Nintendo could not immediately comment on the project.
Sony has officially endorsed Linux efforts for its market-leading PlayStation 2 console, going so far as to offer a $199 kit on its Web site with a keyboard, hard drive, networked adapter and software to turn any PS2 into a Linux computer.
The Linux software system, which can be freely modified by users, is an increasingly important rival to software made by Microsoft, particularly in corporate applications, such as running servers.