Yes, many insurance companies will allow you to start a policy with a Mexican driver's license. However, you will usually have 30 days to provide a valid US driver's license to the insurance company after the start of the policy in order to keep it valid.
If you do want to get insurance in one of the US states and will be obtaining a state license shortly, start comparing quotes to find the best deal on car insurance
So while it is possible to obtain a car insurance policy with your Mexican driver's license, you will need to acquire a state license to keep the insurance in force. If you will only be in the US for a short period of time and do not plan on acquiring a US license, then you may need to see if your Mexican insurance provider will cover you while in the United States.
If you are bringing a vehicle from Mexico into the US, state law will likely require that you have a current Mexican driver's license, current Mexican license plates and current vehicle registration. If the car isn't registered in your name, get a letter from the owner authorizing you to drive the car.