The development of sensors for monitoring Carbon Monoxide (CO) in a large range of temperature is of crucial importance in areas as monitoring of industrial processes or personal tracking using smart objects. Devices integrating GaN/Ga₂O₃ core/shell nanowires (NWs) is a promising solution allowing combining the high sensitivity of the electronic properties to the states of GaN-core surface; and the high sensitivity to CO of Ga₂O₃-shell. Because the performances of sensors primarily depend on the material properties composing the active layer of the device, it is essential to control these properties and in first time its synthesis. In this work, we investigate the synthesis of GaN/Ga₂O₃ core-shell NWs with a special focus on the formation of the shell. The GaN NWs grown by Plasma-assisted molecular beam epitaxy, are post-treated following thermal oxidation to form Ga₂O₃-shell surrounding the GaN-core. We establish that the Ga₂O₃-shell thickness can be modulated from 1 up to 14 nm by changing the oxidation conditions, and follows the diffuse-controlled reaction. By combining XRD-STEM and EDX analysis, we also demonstrate that the oxide shell formed by thermal oxidation is crystalline and presents the β-Ga₂O₃ crystalline phase, and is synthesized in epitaxial relationship with the GaN-core.