In a vehicle cooling system, a quantity of warm coolant stored within a thermally insulated tank is used to heat engine lubricating oil in an engine warm-up phase following a cold start. A conduit feeding coolant leaving the engine is connected to an inlet of the tank via a reduced cross-section or a labyrinth pathway. The conduit is connected to an inlet of an electronically controlled distribution valve having three outlets connected to the oil cooler, a passenger compartment heater, and a radiator. In an initial part of the warm-up phase, the valve is closed, and the entire flow of coolant leaving the engine flows into the tank, moving the quantity of warm coolant previously stored in the tank to the oil cooler, where it contributes to more rapid heating of the lubricating oil. When the engine is switched-off, the tank is again filled with warm coolant from the engine.