Assange is in no "immediate danger", it said in a statement, and pointed out that he is the subject of a European arrest warrant issued by Sweden for rape allegations dating back to 2010, although three of the four charges have been dropped.
Assange fears that he will be extradited to the US if he leaves the Ecuadorian embassy.
Assange's letter was not a formal asylum request, which would lead to a long-drawn-out procedure but in it he tells Hollande that "only France" can provide him the protection he needs.
He claims his life is in danger, as well as his physical and psychological well-being.
"By welcoming me, France would carry out a humanitarian and symbolic gesture, sending encouragement to every journalist and whistleblower," he adds.
Wikileaks last month revealed that the US's NSA had spied on Hollande and other leading French figures.
Although Justice Minister Christiane Taubira said in June that she would not be shocked if France granted Assange and American whistleblower Edward Snowden asylum, Prime Minister Manuel Valls has ruled it out on more than one occasion.