“Over the last week, photographers from Voici magazine scaled our fence, climbed our tree and illegally took pictures of our infants inside our home,” Clooney said in a statement.
“Make no mistake, the photographers, the agency and the magazine will be prosecuted to the full extent of the law. The safety of our children demands it.”
Paparazzi photos of the babies on the cover of the latest Voici issue show one with each parent apparently at their home in Italy.
The twins, Ella and Alexander, were born in London in June.
The actor-director, 56, and wife Amal Clooney, a human rights attorney, 39, wed in 2014.