A federal jury will hear opening arguments Sept. 4 in a high-stakes lawsuit that accuses State Farm of secretly bankrolling the campaign of an Illinois Supreme Court Justice who then cast the deciding vote to overturn a billion-dollar judgment against the insurer.

The lawsuit pits State Farm Mutual Automobile Insurance Company against a group of policyholders who say the insurance giant broke racketeering laws by secretly funding the winner of the most expensive state supreme court campaign in U.S. history—and...