We investigate by density-functional theory simulations several elementary reactions associated to direct methane-to-methanol conversion on clean Rh(111) surfaces and on Rh adatoms on Rh(111). Energy barriers and reaction paths have been determined by the nudged elastic band method. The rate-limiting step in the process, C-O bond formation, has higher activation energy than the one for complete methane dehydrogenation. Our analysis enables us to understand the effect of defects on the reactivity and rules out Rh as candidate catalyst for methanol synthesis.