Selective dissolution of a tungsten (85 wt.%)–rhenium (15 wt.%) alloy with rhenium in hydrochloric acid at the temperature of 298 K and anodic polarization modes was carried out to develop a porous catalytic substrate and to recycle rare metals. The parameters of the effective selective anodic dissolution of the tungsten–rhenium alloy, including the differences in applied potentials and electrolyte composition, were found. It was established that samples of the tungsten–rhenium alloy possess the smallest average pore size after being exposed for 6000 s. The obtained porous tungsten samples were characterized by X-ray diffraction and scanning electron spectroscopy. A thermodynamic description of the processes occurring during the anodic selective dissolution of a binary alloy was proposed. In the course of the work, the selectivity coefficient was determined using an X-ray fluorescence wave-dispersion spectrometer XRF-1800. The existence of a bimodal structure on the tungsten surface after dealloying was proved.