SOFIA, Bulgaria — Bulgaria insisted Tuesday that Russian aircraft had violated international rules while flying over the Black Sea.

The Bulgarian Ministry of Defense said Russian aircraft turned off their transponders — communication devices that allow an airplane to be located — and flew without previously announced flight schedules.

Bulgaria, a NATO member, was reacting to a statement on Monday by a spokesman for Russia’s Defense Ministry who denied any such violations. He said Russian aircraft followed international rules and always kept their transponders on.

The exchange of accusations started on Sunday, when Bulgarian Defense Minister Nikolay Nenchev said in an interview with private Nova TV that there had been a rise in violations of Bulgarian airspace by Russian military and commercial aircraft.

Nenchev said Russian military aircraft had violated ‘‘Bulgaria’s area of responsibility’’ in NATO airspace four times in the past month.

‘‘It is very worrying, so we have to take preventive measures’’ scrambling Bulgarian fighter jets, he said.

Nenchev said Bulgaria had demanded an explanation from Moscow, which has insisted it has not received any official inquiries.

Associated Press