Gaymard, the wife of Hervé Gaymard, former minister of Economy, Finance and Industry in the government of Jean-Pierre Raffarin, reportedly received several threatening emails to her work email account at the end of October.
In the messages, a mysterious correspondent demands to be sent confidential documents and contacts from Gaymard's address book.
The blackmailer then threatened to disseminate “personal documents” if Ms Gaymard refused to comply.
The Paris prosecutor's office has tasked a police unit with investigating the emails.
Ms Gaymard has said she does not wish to comment on the case.