Context. Methanol is formed via surface reactions on icy dust grains. Methanol is also detected in the gas-phase at temperatures below its thermal desorption temperature and at levels higher than can be explained by pure gas-phase chemistry. The process that controls the transition from solid state to gas-phase methanol in cold environments is not understood. Aims. The goal of this work is to investigate whether thermal CO desorption provides an indirect pathway for methanol to co-desorb at low temperatures. Methods. Mixed CH₃OH:CO/CH₄ ices were heated under ultra-high vacuum conditions and ice contents are traced using RAIRS (reflection absorption IR spectroscopy), while desorbing species were detected mass spectrometrically. An updated gas-grain chemical network was used to test the impact of the results of these experiments. The physical model used is applicable for TW Hya, a protoplanetary disk in which cold gas-phase methanol has recently been detected. Results. Methanol release together with thermal CO desorption is found to be an ineffective process in the experiments, resulting in an upper limit of ≤ 7.3 × 10⁻⁷ CH₃OH molecules per CO molecule over all ice mixtures considered. Chemical modelling based on the upper limits shows that co-desorption rates as low as 10⁻⁶ CH₃OH molecules per CO molecule are high enough to release substantial amounts of methanol to the gas-phase at and around the location of the CO thermal desorption front in a protoplanetary disk. The impact of thermal co-desorption of CH₃OH with CO as a grain-gas bridge mechanism is compared with that of UV induced photodesorption and chemisorption.