Insurers tell us drivers should notify them of pothole damage immediately, regardless of whether they intend to claim. You'll then usually have around five or six months before you have to file an insurance claim, which gives you time to see if a claim to the responsible authority will work first.
Be aware if you do notify your insurer, there might be a risk in some cases of your premium being affected for the following year, even if you don't actually claim. This is impossible to quantify (plus big insurers have told us claiming WOULDN'T affect next year's premium), but it's something you should consider.