Goa travel Consultancy (+ 91 9486193977 or +91 9423882947)offers both AC and Non AC cars on hire. Car rental charges for basic Maruti Van car is Rs.900 per day ; Rs. 6300/- per week. Fuel charges are not included in the rent.
I do know many tourists who have hired self driven cars from this operator. You will have to pay a refundable security deposit of Rs.5,000/-. They deliver the car free of charge within Panaji city limits and at Airport. They could charge Rs.400/- extra to deliver the car at South Goa.
Another provider is Amit Sangelkar (09822584787)who provides Scorpio on hire for Rs.1,500. I have seen this guy's ads in classifieds but do not know him personally.
FYI, Hiring bikes at Goa works out cheaper @ Rs.300/- per day