The whistleblower, who has been living in Russia to escape US prosecution, had said he would “love to see” Emmanuel Macron allow him to live in France.
He unsuccessfully applied in 2013 and asked again this week, arguing that “protecting whistleblowers is not a hostile act”.
He initially fled to Hong Kong before settling in Russia.
Foreign Minister Jean-Yves Le Drian told CNews television that when Mr Snowden first asked for French asylum in 2013 the government “considered it was not the time. I don’t see what has changed”.
Mr Snowden has also sought asylum in several other countries.
The former NSA contractor said this week that he would like to return to the United States if guaranteed a fair trial.
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“I am not asking for a parade. I’m not asking for a pardon,” he said in an interview. “What I’m asking for is a fair trial.”
He requested the jury to consider the merits of his leak, which revealed mass government surveillance taking place in the US and abroad, rather than simply whether he broke the law.
Moscow has extended his asylum to 2020, although the whisteblower has been an open critic of the government.
He said last year that he “strongly” disagreed with president Vladimir Putin’s policies and called the government “corrupt in many ways”.
His memoir was released in about 20 countries this week, including France.
The United States has filed a lawsuit against Mr Snowden, saying the memoir violates non-disclosure agreements..
Additional reporting by Associated Press