Getty Images/Justin Sullivan
Such a move could put the company in direct competition with on-demand car service Uber, which has also hinted at plans to offer an autonomous taxi service.
Uber executives have reportedly seen screenshots of what appears to be Google's ride-hailing app, according to Stone's sources, and are "deeply concerned" about the possibility of Google using its self-driving cars to pick up and drop off passengers.
To make matters more interesting, Google's venture arm, Google Ventures, has invested $258 million in Uber. Google's chief legal officer, David Drummond, has been on the company's board of directors since 2013.
Drummond reportedly informed Uber's board of the possibility that Google could launch its competitive service, and the board is weighing whether he will be asked to resign.
Stone's report comes hot on the heels of a TechCrunch report revealing Uber's plans to start working on its autonomous vehicles at a research facility in Pittsburgh.
If both reports prove accurate, Uber and Google could become intense rivals. Google estimates that its driverless-car technology will be ready for widespread use in two to five years. Even if Uber does start cranking away at its own autonomous vehicle, Google has a 10-year head start. (Sebastian Thrun, who heads Google's initiatives, won a DARPA challenge back in 2005 for his team's robotic car.)
Right now, Uber uses Google Maps data to power its apps for drivers and riders. That relationship could come under strain if Google has ride-hailing ambitions.