Google is gearing up for an expansion of its New York City real estate that could add space for more than 12,000 new workers, an amount nearly double the search giant’s current staffing in the city, according to people familiar with the matter.

The plan, which hasn’t been previously disclosed, would give Google room for nearly 20,000 staff in the city, including those it has now—rivaling the approximately 25,000 jobs Amazon.com Inc. is projected to add if it completes plans for a major new office in New York.

The...