People are being advised to only travel if it is essential and to not attempt to drive through flood water.
A yellow weather warning has been issued by the Met Office as a result of the storm.
A spokesperson for the force said: “With the arrival of Storm Darragh we are expecting flooding and road closures in some areas.
“Please only make essential journeys and take extra caution on the roads. Do not attempt to drive through flood water. You are putting yourselves and others in danger and it could damage your vehicle.
“Your local councils – Wiltshire Council and Swindon Borough Council – will have more information about road closures.”