We investigate the wake structure and the three-dimensional stability of the two-dimensional von Karman vortex street developing in the wake of a permeable circular cylinder. The flow through the porous medium, assumed homogenous and isotropic, is described by the Darcy law, with a Navier slip coupling condition at the interface with the pure fluid region. The two-dimensional and steady flow past the cylinder is initially considered. Permeability induces a downstream displacement of the recirculation region, which reduces its dimensions until it eventually disappears. Linear stability analysis shows that the flow is progressively stabilized as permeability increases. We identify a critical value of permeability beyond which the steady wake is linearly stable independently of the Reynolds number. Two-dimensional, time-dependent simulations are then carried out. A progressive downstream displacement of the region of onset of the vortex shedding is observed, together with a decrease in the oscillation frequency. Oscillations of aerodynamic forces are progressively quenched with permeability owing to the downstream displacement of the onset region of the vortex shedding. At the same time, traveling vortices are observed far downstream of the body, in opposition with the impervious case, characterized instead by the formation of two shear layers of opposite vorticity, at very large distances from the body. We perform linearized simulations for the evolution of three-dimensional perturbations on the two-dimensional von Karman vortex street. The growth rate and the spatial structure of the perturbations are extracted from such linearized dynamics by employing a sparsity-promoting dynamic mode decomposition (SP-DMD). As permeability increases, the unsteady vortex street past the cylinder is progressively stabilized with respect to three-dimensional perturbations until the transition to three-dimensionality is prevented. We identify a critical value of the permeability beyond which the vortex shedding preserves its two-dimensionality, at least in the considered parameters space.