MOSCOW—Alarmed by Russia’s growing partnership with China, Beijing’s Asian neighbors are increasingly looking for ways to court Moscow away.

That is exactly what Indian Prime Minister Narendra Modi and Japanese Prime Minister Shinzo Abe aim to do at the annual Eastern Economic Forum in Vladivostok this week. Chinese leader Xi Jinping will be notably absent from the event, Russian President Vladimir Putin’s showcase for investment opportunities in his country’s Far East.

“Both...