Julie Gayet had filed a complaint against “persons unknown”, a common formulation under French law, on 18 March over rumours circulating on Twitter and blog sites for some weeks, the prosecutor’s office said.
Mr Hollande and first lady Valerie Trierweiler are unmarried but have been in a relationship for several years. They met when she was working as a journalist for a French magazine.
Their relationship came under media scrutiny during Mr Hollande’s campaign for the May presidential election and in the months afterwards as Ms Trierweiler became an active tweeter and maintained a literary review column in Paris-Match magazine.
Ms Gayet, 40, has dozens of film credits to her name including a minor role in the 1993 Franco-Polish drama Three Colours: Blue and a leading role in Select Hotel, for which she won two awards for her depiction of a young drug addict.