A normothermic perfusion device to maintain in optimal physiological conditions a liver or kidney intended to be transplanted, wherein said device comprises a permanent part and a disposable part, wherein: - the permanent part comprises: - pumps, heating elements and a control system, configured to regulate fluids in a hydraulic circuit and the temperature of said fluids, whether the container houses a liver or a kidney, and - the disposable part comprises: - a container that in turn comprises: - a hydraulic circuit comprising one first tube that channels a perfusion fluid externally pumped to the container to the liver or kidney housed in a central portion inside the container, and a second tube that channels a vasodilator fluid, - a cannula that channels urine or bile outward from the liver or kidney, respectively, to an external reservoir through a first outlet opening in the container, characterized in that the central portion comprises a lower section which has a height difference with respect to an upper section and the liver or kidney is arranged between said upper section and the lower section, so that the perfusion fluid after circulating through the liver or kidney falls into the lower section, said lower section thus acting as a reservoir for perfusion fluid, and the perfusion device further comprises a connection system connected right at the inlet of the artery of the liver or kidney and fluidly communicated to the first tube and to the second tube, thus enabling the vasodilator fluid to be introduced directly into the artery. - the permanent part comprises: - pumps, heating elements and a control system, configured to regulate fluids in a hydraulic circuit and the temperature of said fluids, whether the container houses a liver or a kidney, and - pumps, heating elements and a control system, configured to regulate fluids in a hydraulic circuit and the temperature of said fluids, whether the container houses a liver or a kidney, and - the disposable part comprises: - a container that in turn comprises: - a hydraulic circuit comprising one first tube that channels a perfusion fluid externally pumped to the container to the liver or kidney housed in a central portion inside the container, and a second tube that channels a vasodilator fluid, - a cannula that channels urine or bile outward from the liver or kidney, respectively, to an external reservoir through a first outlet opening in the container, - a container that in turn comprises: - a hydraulic circuit comprising one first tube that channels a perfusion fluid externally pumped to the container to the liver or kidney housed in a central portion inside the container, and a second tube that channels a vasodilator fluid, - a cannula that channels urine or bile outward from the liver or kidney, respectively, to an external reservoir through a first outlet opening in the container, - a hydraulic circuit comprising one first tube that channels a perfusion fluid externally pumped to the container to the liver or kidney housed in a central portion inside the container, and a second tube that channels a vasodilator fluid, - a cannula that channels urine or bile outward from the liver or kidney, respectively, to an external reservoir through a first outlet opening in the container,