SYDNEY—An Australian program to let foreigners effectively buy permanent residency has stumbled on a problem: Rich Chinese are proving reluctant venture capitalists.

Australia last year revamped its millionaire visa program—officially, the Significant Investor Visa program—which offers a fast track to residency rights in return for investment. With an eye to rebalancing the economy as a long mining boom ends, the new rules call for the required five million Australian dollars (US$3.9 million) to go into growth assets, including...