In a statement on January 4, Bardot says authorities have ignored her "numerous proposals" to save Baby and Nepal, a pair of elephants suffering from tuberculosis.
The animals were ordered to be put down by a court in the southern city of Lyon as a precautionary measure.
The owners of the circus said they would appeal to save them.
Bardot says that if the elephants are killed she will request Russian citizenship "to flee this country that is now just a graveyard for animals."
Bardot's threat comes after fellow actor Depardieu became a Russian citizen in protest at tax hikes.
-- Based on reporting by AP and Reuters