The 'Fashion Police' co-host is eager to take responsibility for the late star's rescue puppies, a Havanese, a Japanese Chin, a Jack Russell terrier and a border collie, reported Contactmusic.
"They were so close and really bonded over their love of dogs. Kelly feels caring for them would be a way to keep Joan close," a source said.
Rivers' daughter Melissa is apparently so busy sorting out her mother's estate that she is thinking about giving the dogs to Osbourne, 29.
"If Melissa's son, Cooper, says he wants the dogs, she'll keep them... But she knows they'd have a good home with Kelly. Kelly already has dogs and knows she could care for Joan's as well," the source added.