Average matrix elements of ordinary muon capture (OMC) to the intermediate nuclei of neutrinoless double beta (0νββ) decays of current experimental interest are computed and compared with the corresponding energy and multipole decompositions of 0νββ-decay nuclear matrix elements (NMEs). The present OMC computations are performed using the Morita-Fujii formalism by extending the original formalism beyond the leading order. The 0νββ NMEs include the appropriate short-range correlations, nuclear form factors, and higher-order nucleonic weak currents. The nuclear wave functions are obtained in extended no-core single-particle model spaces using the spherical version of the proton-neutron quasiparticle random-phase approximation with two-nucleon interactions based on the Bonn one-boson-exchange G matrix. Both the OMC and 0νββ processes involve 100-MeV-range momentum exchanges and hence similarities could be expected for both processes in the feeding of the 0νββ intermediate states. These similarities may help improve the accuracy of the 0νββ NME calculations by using the data from the currently planned OMC experiments.