The report, aired over the weekend, mentioned the case of a young woman whose mother said she had been the victim of an enforced disappearance.
The woman later appeared on a local television show saying she had run away from her mother, married, and had a child.
The prosecution said in a statement its head had tasked lawyers with monitoring outlets and social media "in light of recent observed attempts to harm the security and safety of the homeland by publishing lies and false news."
Egypt's State Information Service, which regulates foreign media in the country, has called on the BBC to retract its story or face a government boycott.
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A BBC spokesperson said that "we are aware of the reports about this BBC story on Egyptian TV and of the comments of the head of the State Information Service. We stand by the integrity of our reporting teams."
The State Information Service has repeatedly complained about media using anonymous sources.
An Egyptian anti-terrorism law criminalises the publication of tolls relating to militant attacks that contradict official numbers.
Egypt is currently gearing up for presidential elections on March 26-28 that President Abdel Fattah al-Sisi looks certain to win.