That sounds consistent with a Jacobs taper.
The chuck has a hole all the way through the whole is tapered to accept a Jacobs taper.
If you Google chuck Arbor you will see what I mean but I doubt it has the second taper to fit the machine (such as a morse taper)
I think it sounds like I know what I am talking about but I might not be.
I think you're onto something here! I googled 'chuck arbor' as you suggested, and they do have a little hole in the top, which is what I can see when I look inside the chuck on my drill.