French election rules state that no opinion polls or predictions can be published between midnight on Friday before the election day (always a Sunday) and the close of polling.
Media organisations are banned from publishing the results of any polls and risk fines of between €3,500 and €75,000 ($4,500 and $100,000).
Twitter has around 5.2 million users in France and Facebook has many more.
If enough users start to spread news of exit polls on voting day, authorities are worried that they could start to influence voting.
"Sending a tweet or a message on Facebook fall under the law," said Mattias Guyomar, general secretary of the French Polling Commission, reported Sunday's Le Journal du Dimanche.
The newspaper reported that legal experts are worried that the spreading of information could even lead to "the cancellation of the election."
Polling organisations use exit poll data to create predictions of the outcome. These tend to be circulating in news rooms by early evening, while polls close at 8pm.
This year, there is a real risk that the "happy few" of those in the know "will be much longer" said Mickaël Darmon, a political commentator with TV channel i>Télé.
Problems could occur in marginal areas where knowledge of candidate support might affect how those voting later cast their ballots.
"In the case of an extremely close result between two candidates, it would be within the loser's rights to appeal the results before the Constitutional Council and demand that the election be cancelled," reported the newspaper.
The first round of voting will take place on Sunday April 22nd, with a second round for the two best-placed candidates to follow two weeks later on Sunday May 6th.