The boy, named locally as Thomas Gacek, was taken to Royal Oldham Hospital last Saturday with serious head injuries.
He was transferred to The Royal Manchester Children's Hospital where he died on Tuesday.
The mother's 35-year-old boyfriend and the their lodger, a 26-year-old man, have also been arrested, reports Sky News.
Their house in the Clarksfield area of Oldham is being forensically examined.
Police told the Manchester Evening News they were a still working to establish the identity of the baby's father.
Det Insp Andy Cunliffe, from GMP's Major Incident Team, said: "This is a tragic case in which a very young baby has lost his life and we are determined to find out exactly what the circumstances were that led to him suffering such injuries.
"News of the death of such a little boy will no doubt come as a great shock to people and I would like to try and reassure the community as much as I can that we have a team of detectives dedicated to working on this investigation."
The boy's mother and her boyfriend were originally arrested for assault and bailed. They were then re-arrested on suspicion of murder and conspiring to pervert the course of justice.