Detections of gas phase methanol CH$_3$OH in the cold regions of protoplanetary disks are still poorly understood. The role of X-rays, emitted from the central young stellar object (YSO), to explain these observations is still an open question. X-ray photodesorption, which is the desorption of molecules from interstellar ices, induced by its X-ray irradiation, is a potential candidate to explain the observed CH$_3$OH abundances. We have experimentally studied this process in the soft X-ray range near the O K-edge ($\sim$ 560 eV) on CO or H$_2$O-rich ices containing methanol. The X-ray photodesorption spectra of the desorbing molecules are found to be well-correlated with the ice X-ray absorption spectrum. X-ray photodesorption yields, which represent the efficiency of a X-ray to desorb a given molecule, are found to be strongly dependent on the ice composition. X-ray photodesorption of CH$_3$OH is efficient (with a yield of $\sim$ 10$^{-2}$ molecule/photon at 565 eV) when CH$_3$OH is embedded in CO-rich ices. When embedded in H$_2$O-rich ices, the X-ray photodesorption of CH$_3$OH is not detected. This was assumed to be due to the X-ray induced-chemistry. Consequently, X-ray photodesorption should participate into explaining the presence of gas phase CH$_3$OH only beyond the CO snowline of protoplanetary disks.