The youngster, who has not yet been named, was rushed to hospital last Tuesday and died five days later.
Emergency services were called to a property in Cranworth Road, Burngreave, at 6.10pm following reports that the child was having difficulty breathing.
Her death is not being treated as suspicious.
Enquiries into the circumstances surrounding the death are being carried out ahead of an inquest.
A South Yorkshire Police spokeswoman said: "Emergency services were called to a property in Cranworth Road, Sheffield, just before 6.10pm on Tuesday, January 2 after it was reported that an eight-year-old girl was having difficulty breathing.
"She was taken to hospital where she sadly died on Sunday, January 7.
"There are no suspicious circumstances surrounding her death and a file will be prepared for the coroner."