The former Manchester United forward, who would need the backing of 500 elected officials by the end of February to run, has sent a letter to French mayors describing himself as an "engaged citizen" and asking for their support, the left-leaning newspaper said.
"This engagement obliges me to speak, more earnestly than usual, but also with a keen sense of my responsibility, at a time when our country faces difficult choices which will be decisive for its future," said the letter, quoted in Liberation.
In his letter, Cantona denounced the limited opportunities for young people in France and social injustices which were "too numerous, too violent, too systematic," the newspaper said.
Election candidates will compete in a first round in late April, with the two biggest vote-winners going through to a decisive second round in early May.
Socialist candidate Francois Hollande is currently leading President Nicolas Sarkozy in the polls.
Even if Cantona were able to get 500 signatures, a difficult task without the backing of a party, he would have virtually no chance of reaching the second round as an individual outsider.
It is not the first time Cantona, who earned large sums as a footballer from sponsorship by brands like L'Oreal and Nike, has tried to draw public attention to social inequality.
In December 2010, he called on French savers to stage a nationwide bank run by withdrawing their money from financial institutions because of their role in triggering the global financial crisis. Panned by the media as a flop, almost no one in France heeded the much-hyped call.
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