Since it's not entirely sure that we as a whole person have entirely free will (instead of a really elaborate after-the-fact-excuses machine), I'd think the concept of a "will" for individual systems needs to be very loosely defined to apply here.

If you're interested in the science, I'd suggest starting from the Libet experiments (who essentially thinks his experiments show that the subconscious makes decisions, and the conscious retains a veto right)

It's a fascinating thought to entertain. (But you will forever doubt if you our your subconscious made the decision to think about this ;)

If you care more for pop-sci, Kahnemann's "Thinking fast and slow" idea of system1/system2 thinking might be a good read.

That still doesn't mean that different consciousnesses share a mind. We certainly have different systems competing for attention, but what consciousness even means. (I'd recommend Popper as a starting point for that debate, but mostly because the science is still very much in the "Dunno, but what happens if we poke here?" stage, IMHO. Philosophy seems more apt)