The spy agency taped 70.3 million phone calls in France over a 30-day period between December 10 and January 8, 2013, Le Monde reported in its online version.
According to the paper, the NSA automatically picked up communications from certain phone numbers in France and recorded text messages under a programme code-named "US-985D."
Le Monde said the documents gave grounds to think the NSA targeted not only people suspected of being involved in terrorism but also high-profile individuals from the world of business or politics.
US authorities declined comment to the French daily on the "classified" documents.
The Le Monde article followed similar revelations by German weekly Der Spiegel that US agents had hacked into the email account of former Mexican president Felipe Calderon.
Mexican authorities have said they will be seeking answers from US officials "as soon as possible" following the allegations.
Snowden, who has taken refuge in Russia, is wanted in the United States for espionage and other charges after leaking details of the NSA's worldwide snooping activities.
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