Faced with legislation in California that endangers their business model, Uber Technologies Inc. and Lyft Inc. are urging a compromise that would keep their drivers from being considered employees.

“We can make independent work better if we update century-old employment laws,” Uber Chief Executive Officer Dara Khosrowshahi and Lyft co-founders Logan Green and John Zimmer wrote in a rare, jointly written opinion column published June 12 in the San Francisco Chronicle. “Many drivers are offering ideas to improve their experience, and companies like ours have a responsibility to come to the table prepared to do our part.”

The executives’ public...