The effective resonance energy ((E)over bar(r)) values for the Re-185(n,gamma)Re-186 and Re-187(n,gamma)Re-188 reactions were measured by using the cadmium ratio method. Au-197 and Mo-98 isotopes were selected as dual monitors to determine the epithermal neutron spectrum shape factor in the irradiation sites of a 37 GBq Am-241-Be neutron source. The analytical grade rhenium(VII) oxide samples were diluted with Al2O3 powder to reduce neutron self-shielding effects, and they were put into 1-mm thick small cylindrical polytetrafluoroethylene (PTFE) boxes. Samples were irradiated in a Am-241-Be neutron source. The activation produced in the irradiated samples was measured in close-counting geometry on the end cap of a Ge detector in order to obtain better counting statistics. The correction factors for gamma-ray self-absorption (F-s), thermal neutron self-shielding (G(th)), and epithermal neutron self-shielding (G(ep)(i)) effects were determined with suitable approaches. Consequently, the experimental (E)over bar)(r)-values were measured to be 3.49 +/- 0.54 eV for Re-185 and 41.77 +/- 16.79 eV for Re-187 target nuclide, respectively. Furthermore, the (E)over bar)(r)-values were theoretically computed through up-to-date resonance data obtained from the ENDF/B VII library using two different approaches. Since there is no experimental data available in the literature for the (E)over bar)(r)-values of these isotopes, the results were compared with the theoretical values in the literature.