The invention relates to a device (3) for supporting packaging for transporting/storing radioactive materials (1) in a horizontal position, the supporting device including a structural base (10) and structures (12) for supporting the packaging, which structures are supported by the base (10) and project upwards therefrom. According to the invention, the device also includes, located at least in part above the structural base (10), a shroud (30) for guiding air for cooling the packaging by natural convection, the shroud (30) defining a recess (32) which is open towards the top and into which a portion of the packaging is intended to be inserted when the packaging (1) is supported in a horizontal position on the device, the shroud (30) including, at the bottom thereof, at least one opening (34) for allowing cooling air into the recess (32).