"The child remains in hospital and is receiving ongoing treatment.
"The McGowan family asks for privacy regarding this situation at this difficult time."
McGowan is the father to three children - two boys and a girl.
The WA Premier confirmed that he was entering isolation protocols until at least next Thursday.
"It is not surprising considering one of my family members is positive," he said.
"In accordance with protocols, this will extend my quarantine period until I am hopefully cleared to leave home next Thursday afternoon, at the earliest."
It comes after news of opposition leader Anthony Albanese testing COVID-positive, in the midst of Labor's election campaign.
Get the latest breaking news from SCA newsrooms around Australia. Short, simple and everything you need to know.
Download LiSTNR and enjoy a new world of audio. All your favourite shows and stations in one library.