The 36-year-old became the first female coach of a professional club in the top two tiers of any European league when appointed to the position last month
Costa was officially presented to the media as Clermont boss on May 22, becoming the first female coach of a professional club in the top two tiers of any European league.
But in a decision the club has described as "sudden and surprising", the 36-year-old has opted against assuming her position for the coming season. No reason has been made public.
"I deeply regret this situation," Clermont president Claude Michy said in a statement. "I thank all those who have given me their support and I am very appreciative.
"Club operations will continue with other stakeholders in order to prepare for the new season."