The man, 35, and woman, 27, have been arrested on suspicion of manslaughter by gross negligence after the incident in Woodlands on Sunday.
A South Yorkshire Police spokesman said the tot had suffered serious injuries after being attacked by a dog and died a short time later in hospital.
A brief statement said: “Police were called to an address on Welfare Road at around 3.30pm on Sunday following reports that a dog had attacked a child.
"On arrival at the property, emergency services discovered a 12-day-old baby boy had been bitten by a dog and suffered serious injuries. The child was taken to hospital, but sadly died a short time later.
"A 35-year-old man and a 27-year-old woman have been arrested on suspicion of gross negligence manslaughter. They have now been bailed while enquiries continue.”
The dog has been removed from the address.