"This did not happen," Magdy Abdel Ghaffar said at a press conference when a reporter asked if Regeni, a Cambridge University PhD student, had been arrested by the police.
"It is completely unacceptable that such accusations be directed" at the interior ministry, he said.
"This is not Egyptian security policy -- Egyptian security has never been accused of such a matter."
Regeni disappeared on January 25 and was found dead on February 3.
Rights activists and several opposition groups say Regeni, who was doing research on Egyptian trade union movements, had been arrested by the police and tortured.
The diplomatic community and the Italian media have also raised the possibility of torture.
Global rights groups have regularly denounced mysterious disappearances of activists, torture and beating of detainees in Egyptian detention centres.
President Abdel Fattah al-Sisi has himself urged security forces to restrain themselves after several cases of custodial deaths emerged in recent months.
On the anniversary Cairo was quiet, with police deployed across the capital to prevent any demonstrations.
Regeni's body was found in a ditch in a Cairo suburb bearing signs of torture.
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