What you're probably looking for is "magnet wire" and not magnetized wire. Magnet wire was a common term for insulated wire used in winding electromagnets, transformers, motors, inductors, etc. The insulation is most commonly enamel these days, whereas it was cloth and paper in years past.
I would look for 22 gauge enamel-covered solid copper wire. Since it's enamel-covered copper, there's no need for galvanization.
To make an electrical connection to the enamel-covered wire you'll need to burn off the enamel, which is basically an insulating varnish, with a lighter or other flame. Then sand the area with emery cloth or the like until it brightly shines. Clean it with a damp cloth, and you're ready to solder.
