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Rhenium-186 (Re-186) has attracted attention as a medical isotope. The feasibility of producing Re-185, the raw material for Re-186, using a fast reactor was evaluated using a continuous energy Monte Carlo code. The irradiation of natural tantalum (Ta) in the fast reactor can produce Re-185 with an isotopic purity of 99%. A two-step irradiation process with different moderators was found to improve the production rate of Re-185. Specifically, this can be achieved by using zirconium hydride (ZrH_1.7) as a moderator in the first transmutation process from natural Ta to tungsten (W), and then zirconium deuteride (ZrD_1.7) as a moderator in the second transmutation process from W to Re-185. Due to the two-step irradiation, the production rate of Re-185 from Ta can be increased up to a maximum of 470 times compared with irradiation without a moderator, and 2.3 g of Re-185 can be obtained from 1571 g of Ta in 1 year of irradiation. The proposed isotope production method is a new method that is different from the conventional electromagnetic enrichment process.