Leading gig economy companies including Uber and Lyft are quietly lobbying California’s top Democrats to override or undermine a court ruling that could make many of their contract workers into employees.

In April, the California Supreme Court issued a far-reaching ruling which could make it much harder for companies to claim their workforces of independent contractors are not full-fledged employees under the state’s wage laws. Over the months since, business leaders have been pleading their case to state officials including members of Governor Jerry Brown’s cabinet, Brown’s presumed successor Gavin Newsom, and members of the state legislature.

The business leaders...