The newborn boy was found unresponsive at a property in Piccadilly Road, Burnley, at around 12.10pm on November 28, 2021.
The boy was taken to Manchester Children's Hospital for treatment but died two days later.
A 25-year-old man from Burnley was arrested on suspicion of assault before being rearrested on suspicion of murder following the baby's death.
Lancashire Police said a post-mortem examination had been conducted, but further medical investigation was required to establish a cause of death.
A 25-year-old woman, who was also arrested on suspicion of assault and later on suspicion of murder, has been released without charge.
The man has been released on bail until August 31.
Anyone with information can contact the police by calling 101, quoting log reference 0581 of November 28.
Independent charity Crimestoppers can also be contacted anonymously on 0800 555 111 or online at https://crimestoppers-uk.org/.