The elephants, which suffer from tuberculosis, were due to be killed in December, but were granted a reprieve until after Christmas.
Brigitte Bardot said she would move abroad if the reprieve was not made permanent.
Fellow actor Gerard Depardieu obtained a Russian passport this week following a tax row with the French government.
Gerard Depardieu had earlier threatened to move to Belgium to avoid higher taxes.
After the government criticized his decision, he declared he would give up his French nationality.
Earlier this week, the Kremlin said President Vladimir Putin had personally signed a decree granting actor Gerard Depardieu Russian citizenship.
Vladimir Putin and Gerard Depardieu are said to be personal friends.
Brigitte Bardot, a prominent animal rights campaigner, warned she would follow Gerard Depardieu’s example if authorities failed to save the elephants, known as Baby and Nepal.
“If the powers that be have the cowardice and the shamelessness to kill Baby and Nepal… I have decided to apply for Russian citizenship and leave this country, which is nothing more than an animal cemetery,” Brigitte Bardot said in a statement posted on the website of her animal welfare charity.
The case of the sick elephants, in Lyon’s Tete d’Or zoo, has gripped the city since mid-December.
The authorities deemed Baby and Nepal a health threat to other zoo animals and visitors, and ordered them to be put down.
But a circus director who used to own the elephants launched a petition to save their lives.
Thousands signed the petition, gaining the animals a temporary reprieve.
Brigitte Bardot joined the campaign and wrote a letter to French President Francois Hollande earlier this week, asking him to intervene.
She posted Friday’s announcement after her plea failed to elicit a reply.
Brigitte Bardot has been making headlines in recent years for her controversial remarks about Islam and its followers.
In 2008, Brigitte Bardot was convicted for the fifth time for inciting racial hatred after complaining on her website that Muslims were “destroying our country by imposing their ways”.
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