The partial oxidation of methane to methanol with molecular O2 at mild reaction conditions is a challenging process, which is efficiently catalyzed in nature by enzymes. As an alternative to the extensively studied Cu-exchanged zeolites, small copper clusters composed by just a few atoms appear as potential specific catalysts for this transformation. Following previous work in our group that established that the reactivity of oxygen atoms adsorbed on copper clusters is closely linked to cluster size and morphology, we explore by means of DFT calculations the ability of bidimensional (2D) and three-dimensional (3D) Cu5 and Cu7 clusters to oxidize partially methane to methanol. A highly selective Eley-Rideal pathway involving homolytic C-H bond dissociation and a non-adsorbed radical-like methyl intermediate is favored when bicoordinated oxygen atoms, preferentially stabilized at the edges of 2D clusters, are available. Cluster morphology arises as a key parameter determining the nature and reactivity of adsorbed oxygen atoms, opening the possibility to design efficient catalysts for partial methane oxidation based on copper clusters.