Building on the idea of Adam Zuckerman regarding casing of sorts, I would like to draw your attention to earbud cases (as from a google search). These has loads of variations, but most of them should solve your problem.
Some manufacturers sell the earbuds with decent casings, and it is worth it to look for one of those sets. Cases comes in multiple variants; closed cases, case where you twist the leads into the case, or open variants where the headphone wires are still somewhat available to the cat.
Note that also ordinary headphones can come with decent casings, and some of these headphones are collapsable before you enclose them in the casing.