Greene, 26, was not home at the time of the blaze.
No humans were seriously hurt in the fire and Greene's second pup survived.
"She loves her dogs. LOVES," a source told UsMagazine.com. "She must be absolutely devastated."
There was no word on the cause of the fire.
Greene is best known for her work in the "Twilight" film franchise.
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'
Gal Gadot cast as Wonder Woman for 'Batman vs. Superman'