Mostafa Suleiman told a press conference that Italian investigators had made the demand during an inconclusive meeting in Rome last week that prompted Italy on Friday to recall its ambassador from Cairo.
Regeni, a 28-year-old Cambridge University PhD student, was in Egypt researching labour unions when he disappeared on January 25.
His badly mutilated body was found more than a week later on the side of a road.
Suleiman said Italian investigators asked for records of "all subscribers in areas in where (Regeni) lived, where he disappeared and where his body was found", Suleiman said, adding the number could even reach a million.
Also Read
Suleiman added that the Italian investigators "conditioned further judicial cooperation on this demand" but the Egyptian delegation in Rome flatly refused.
Rome announced it was recalling its ambassador over lack of progress in the probe into Regeni's brutal murder.
Suleiman said that the Italian investigators also demanded CCTV footage that had been automatically deleted by then, but Egypt made inquiries and found that a program could be purchased that might have retrieved it.
He said they asked Italy for help but the matter was "still under study".