Officers were called to a report from the hospital of concern for a baby just before 6.30pm on Friday 8 January 2021.
A two-week-old baby girl was taken to hospital where she sadly passed away on Monday 11 January.
A 24-year-old man has been arrested on suspicion of murder and has since been released on bail pending further enquiries.
An investigation is ongoing and a scene is expected to remain in place at an address in Heywood for the coming days.
A post-mortem is due to take place this week.
Detective Chief Inspector Carl Jones, of GMP’s Major Incident Team, said: “This is a desperately sad incident where a baby girl has lost her life and we are doing what we can to support her loved ones at this incredibly difficult time.
“Our investigation is ongoing; as always, we are keeping an open mind and have specialist officers working at the scene and on the investigation to establish the full facts of this case.
“This is a heart-breaking time and we ask the public to refrain from speculation and respect the family’s privacy at this time.
“Anyone with information should do the right thing and contact us with information so we can continue to try and ascertain the circumstances behind this tragedy.”
Anyone with information should call police on 0161 856 3400 quoting incident 1896 of 08/01/2021. Details can be passed anonymously to the independent Crimestoppers on 0800 555 111.