Using as a template the crystal structure of COVID-19 protease, we developed a pharmacophore of functional centers of the protease inhibitor-binding pocket. Then we conducted data mining of the conformational database of FDA-approved drugs. This search brought 64 compounds that can be potential inhibitors of COVID-19 protease. The conformations of these compounds undergone 3D fingerprint similarity clusterization. Then we conducted docking of possible conformers of these drugs to the binding pocket of protease. We also conducted the same docking of random compounds. Free energies of the docking interaction for the selected compounds were clearly lower than random compounds. Three of the selected compounds were carfilzomib, cyclosporine A, and azithromycin—the drugs that already are tested for COVID-19 treatment. Among the selected compounds are two HIV protease inhibitors and two hepatitis C protease inhibitors. We recommend testing of the selected compounds for treatment of COVID-19.