The 13-year-old, identified as Soher Ebrahim, died on Thursday evening in a village in the Daqahliya governorate, Gulf News reports.
It adds her family have filed an official complaint, accusing the doctor who performed the operation of causing her death.
Soher’s father Mohammad told the Egyptian daily newspaper Al Masry Al Youm his daughter had been one of four girls being circumcised at the clinic at the time.
He said the family had been informed Soher died of an anaesthesia overdose, though this has not been confirmed.
He added: “If I had known the operation was going to kill her, I would never have [allowed] her to have it. The same doctor conducted a similar operation on her elder sister two years ago and villagers use this doctor because he has a remedy for everything at low prices."
Soher’s uncle told the newspaper the doctor in question had offered her family 20,000 Egyptian pounds (around GBP1,850) if they did not file a complaint against him.
Al Arabiya reports the police have ordered an autopsy and have “summoned” the doctor to determine the cause of the girl’s death.
It cites a health inspector report as attributing the cause of death to “a sharp drop in blood pressure resulting from shock trauma.”
The procedure of FGM was criminalised by the Egyptian parliament in June 2008, with those found guilty liable for fines and prison sentences of up to two years.
According to Egypt Independent, recent evidence suggests that younger generations are challenging the practice.
It quotes a 2008 Demographic and Health Survey of Egypt, which recorded that 91.1 percent of women aged 15-49 years underwent FGM, but only around 74 percent of girls aged 15-17 years, a percentage that is expected to drop to 45 percent over the next 10 years.
Compared to results from the 1995 survey, which recorded 96 percent of women aged 15-30 years having experienced FGM, "this is positive - if patient - progress".
The World Health Organisation defines FGM as including procedures that intentionally alter or cause injury to the female genital organs for non-medical reasons.
The procedure has no health benefits and can cause severe bleeding and difficulty urinating. Subsequent cysts, infections, infertility and complications in childbirth can lead to an increased risk of newborn deaths.
About 140 million girls and women worldwide are currently living with the consequences of FGM.