The nine-month-old pugs were stolen last Thursday between 6pm and 9pm. Oscar was back in the garden at 5am the next day, but George, who is black, is still missing.
Oscar's broken leg has been put in plaster with the message "Help find George" scribbled on.
The family have offered a reward for George's safe return, and have spent every day scouring the countryside near their home.
They have gone to St Helens and Wigan putting up posters appealing for the dog's safe return.
. Call the Garnett family on 01744 885915 or Crimestoppers on 0800 555111.