Denise Jarrett, 68, claims she developed cancer as a result of being exposed to asbestos dust and fibre while she worked for the council at the school.
Papers released by the court indicate that the Council has already admitted liability.
Ms Jarrett, now of Kentisbeare, Cullompton, Devon, took up the post as head at Chichester Nursery School in Woodlands Lane in 1997.
Those premises were later demolished and the school, along with Ms Jarrett, moved to different premises in St James Road.
She is asking the court to hear her case urgently because of her limited life expectancy.
The court documents say she is suffering from epitheliod mesothelioma after being exposed to asbestos when she worked for the county council between 1997 and 2005.
Epitheliod mesothelioma can affect the linings of the lungs, heart or abdomen, and the average life expectancy of sufferers is said to be between 12 and 27 months.
The document say she was exposed to asbestos from an asbestos cement pipe just outside her office door and that she used to hang displays or grow plants up.
She also visited the school’s first floor which initially was thick with dust regularly, and the writ says a survey showed asbestos was present in 14 rooms.
The state of it is described as ‘fair’, where it was indented or cracked, and sometimes ‘poor’ where some material had become detached.
Workmen came during school holidays to carry out maintenance work, which probably involved disturbing asbestos, and she was present when the school was finally demolished, emitting asbestos fibres into the atmosphere, she claims.
Ms Jarrett accuses the council, among other things, of negligently failing to avoid her exposure to asbestos, failing to remove all asbestos from the school, and failing to warn her of the asbestos.
The writ claims the council also negligently failed to close the school until it had been made safe, allowed her to go into the buildings during deconstruction which increased her risk of inhaling asbestos, failed to take steps to remove asbestos fibres from the air of the premises, and failed to take any care for her safety.
A West Sussex County Council spokesperson said: “When an incident such as the tragic case of Ms Jarrett arises, and they are thankfully quite rare, we will accept our responsibilities where that is the right thing to do.
“We must then bring in our insurers to help us reach a fair and reasonable settlement that recognises the gravity of what has happened for the family and the extent of our responsibility as the employer and education authority.”