The source also said Uber's board was weighing whether to ask Drummond to resign, Bloomberg reported.
Neither Uber nor Google were available for comment.
Uber executives have been made privy to screenshots of Google's ride-sharing app, which is currently being used by Google's employees, Bloomberg reported.
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Google's move may imperil Uber's reliance on the search giant, as the cab service's smartphone applications for drivers and riders are based on Google Maps.
Google recently said its driverless car technology was in development within its Google X research lab, but it is two to five years from being ready for widespread use, according to Bloomberg.
Uber also announced a tie-up with Carnegie Mellon University on Monday to create the Uber Advanced Technologies Center in Pittsburgh aimed at developing Uber's own self-driving vehicle technology.
The partnership seeks to provide "research and development, primarily in the areas of mapping and vehicle safety and autonomy technology," Uber said in its blog.