There’s been a lot of speculation about what a re-designed iPod nano might look like in the wake of a note issued Thursday by Kaufman Bros.’ Shaw Wu, but what about the software that runs it?

Wu’s sources in Apple’s (AAPL) supply chain turned up plans to buy components for a smaller-size nano with no clickwheel. “We frankly are not sure if this model will see the light of day,” he wrote, describing the design as a “big departure” that might “jumpstart” sales of what was once Apple’s best-selling iPod.

Following up on Wu’s report, iLounge turned up photos of square-shaped cases that would fit such a device. And several sites remembered the photo — since removed under duress from its original Asian source — of what resembled a 3X3 cm Apple-branded touchscreen, just the right size to fit into one of those cases.

But the question nobody seems to be asking is what operating system a touchscreen iPod nano would run. With the exception of the iPod touch, which runs the same iOS as the iPhone and the iPad, all iPods run a miniature version of Mac OS behind a thin layer of imaging software provided by a company called Pixo (see here.)

But if there is a new touchscreen iPod nano coming next week, it will almost certainly need iOS to run the screen. And that would bring it into the iPhone family and, in theory, give it access to some of the nearly 250,000 apps written for the iPhone, iPad and iPod touch.

Talk about a jumpstart!

[Follow Philip Elmer-DeWitt on Twitter @philiped]