A Japanese Foreign Ministry spokesman said the plant will reprocess between 5,000 and 7,000 tons of radioactive waste a year.
The building costs will be drawn from a $100 million aid package Japan set up last November with Russia to help dismantle nuclear weapons held in the former Soviet Union.
The spokesman said Japan did not expect building costs to exceed the fund and hoped to have the results of the bid by the end of next month or early December.
The storage and treatment plant will be placed on a barge and will be moored at a shipyard used for decommissioning nuclear-powered submarines near the Far Eastern port of Vladivostok, the foreign ministry said in a statement.
Russia dumped about 800 tons of low-level nuclear waste in the Sea of Japan last October provoking an outcry in Japan and South Korea.
The move prompted Japan to provide assistance to nations of the former Soviet Union to dismantle unnecessary nuclear weapons.
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