This is so simple that others may have also come up with it. If so, sorry to duplicate. I have an old Bernina 830 and my mom's even older Elna Supermatic. Both have upright spindles. If you have a machine with horizontal spindles where the thread is not supposed to turn, I'm not sure how this would work.
My local store was out of spool adapters when I needed one. In order to use Coats and Clark machine quilting cotton, and other large-hole spools on a small spindle I wrap the bristly side of some self-adhesive Velcro around a piece of tubing that just fits my spindles. It takes a few seconds to make and has worked well for me.
You need just enough Velcro to make a tight but easy fit inside the spool. In this case I went around the tube about 3 times. The Velcro glue side stuck to the bristles better than I expected.
I use a pair of needle nose pliers or tweezers to pull it out of my spools, but if you take the labels off both ends you could just push it through. I had some old tubing around the house, so I'm not sure of the size, but it appears to be 1/4" inch. The inner dimension has to fit well over your spindle, allowing it to turn freely. It's cheap at any hardware store and also sold at pet supply places for use in aquariums. To be sure of the size, you could take an ordinary thread spool to the store with you and find tubing with the same dimension as the hole in the spool. I hope this is a useful.
Here are a couple of pictures. The tube I happened to have on hand is black, but most of it that you find is clear.
Tubing with Velcro wrap
Tube inside the spool