Hannah Black, 13, and 14-year-old Rochelle Cauvet were swept away by a flooded stream at Stainforth Beck, near Settle, as they walked at the edge of the water with a school party in October 2000.
The outing had been part of a residential week organised by Royd's School, in Oulton, near Leeds.
Leeds City Council is facing two charges of breaching health and safety at work regulations following the deaths.
In March, an inquest jury at Harrowgate magistrates' court returned a unanimous verdict of accidental death on both girls.
The Health and Safety Executive said it was prosecuting the council for contravention of the Health and Safety at Work Act 1974.
It claims the council failed to ensure, as far as reasonably practicable, that persons not in their employment, were not exposed to health and safety risks during the walk.
The council is also being prosecuted for failing to complete a suitable and sufficient risk assessment in respect of river walking activities at Stainforth Beck.
The case was adjourned until 3 October at Leeds magistrates' court during a brief hearing.
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