You do not need a police report to open a claim. Since it was a single-vehicle accident, you will be charged as at-fault if the claim is paid, which would result in a higher premium. I would get an estimate for the damage and see if it's worth filing a claim for, and possibly pay out-of-pocket to avoid a rate hike.
The new insurance company is likely to find out about the accident, but that will come from your driving reports when you buy your policy. The majority of insurance companies request a motor vehicle report (MVR) and a comprehensive loss underwriting exchange (CLUE) report to verify the accuracy of a new customer's driving record.
The resulting damage from hitting a pothole is actually considered an "At Fault" accident and would be covered under the collision portion of your policy. The reason this is the case is because insurance companies expect drivers to operate their vehicles safely and watch for hazards in the road to avoid frivolous claims.