This study investigates the principal possibility of a boiling water cooled travelling wave reactor that can potentially improve the natural uranium utilisation in Boiling Water Reactors (BWRs), and more general in water cooled reactors, appreciably. A strong variation of the water density along the core height makes favourable the nuclear fission in the lower core part and the fuel breeding in the upper one. A serial axial fuel shuffling scheme is considered for establishing a kind of travelling wave. The fresh natural uranium oxide fuel is loaded from the core top. The irradiated fuel is discharged from the core bottom. The water coolant at the saturation point at a near atmospheric pressure comes from the bottom of the core, as in conventional BWRs. The asymptotic state of the breeding/burning wave is obtained, including its major characteristics, such as power density distribution, coolant mass flow rate, and the fuel shuffling speed. The employed simulation model for the water density variation in axial direction is based on a slip ratio approximation for two phase flows. The burn up calculations are performed with the ERANOS2.2 code, in which the models of axial fuel shuffling and coolant density variation have been implemented. The numerical results for 1D models are encouraging and show that the breeding performance is sufficient to keep the core critical in the asymptotic state, and a maximum burn up of about 37% can be reached, meaning that the natural uranium utilization in BWR can be improved by several ten times compared to conventional water cooled reactors. Feasibility related issues, such as the void reactivity effect and crit ical heat flux limit, has been checked and discussed. Further efforts are needed to establish a viable design based on the ideas proposed in the paper.