The national financial prosecutor's office (PNF), which specialises in investigating economic crimes, began a preliminary investigation last year.
It is looking into possible offences which include private corruption, conspiracy and influence peddling, the source said.
The source added that former FIFA president Sepp Blatter was interviewed as a witness by Swiss prosecutors in the presence of two French magistrates in Switzerland on April 20.
The 81-year-old Blatter, who has been banned for six years by FIFA's own ethics committee for ethics violations, was not immediately available to comment.
FIFA has been in turmoil since several dozen sports officials, including a number of senior FIFA officials, were indicted in the United States on corruption-related charges in 2015.
In the same year, Swiss authorities opened criminal proceedings against individuals on suspicion of mismanagement and money laundering related to the allocation of the 2018 and 2022 tournaments.
Read the full article on eurosport.co.uk: French prosecutors investigating 2018, 2022 World Cup awards