The wife of Charles' son William was due to deliver their first child July 11.
"We are all just waiting by the telephone," UsMagazine.com said Camilla told the crowd during a visit to Little Harbour children's hospice in Cornwall Monday. "We are hopeful that by the end of the week he or she will be here."
During the visit, Charles happily met a woman and greeted her 3-week-old daughter, the celebrity news website said.
"You have got one of these coming soon," the woman said to the prince, to which Charles replied, "Hopefully!"