The child was born to the same surrogate mother as their son Zachary - who was born on Christmas Day, 2010, the New York Post reports.
John's representative Fran Curtis told the newspaper: "We never discuss our clients' personal lives."
The California-based woman is also thought to have given birth to Camille Grammer's children, but her identity has been kept secret.
John and Furnish were said to be "over the moon" about the upcoming birth of their second child when the news first broke last November.
At that time, the woman reportedly recruited via the Center for Surrogate Parenting in Encino, California, was believed to be "several months" pregnant.
The couple were rumored to have bought a new LA mansion a month earlier to make room for the new addition to their family.
John said last summer that he "definitely" wants another child as a brother or sister to Zachary.
Celebrity pictures - photo gallery
Copyright: WENN Craig Harris/WENN.com