They may be coated with a shellac or varnish insulation. Usually a chemical remover will not work with any speed. It's easiest to just use a sharp knife and scrape the insulation off until you have bare copper wire. Often, simply heating the ends of the wire with a hot soldering iron and solder will melt the insulation off.
It is not likely that you have solid wire for a headphone cable because it would break easily. It may be that you have what is known as Litz wire, which consists of flat bands of copper intertwined with paper. This is also difficult to solder and the copper must be separated from the paper first. It's normally easier to find a tiny crimp connector to do this job.
Dean