Amazon is just one of several potential competitors to established insurance companies. About 60 percent of U.S. consumers are open to buying insurance from new entrants, including startups or companies outside the industry, according to a survey conducted this year by Bain & Company.

"Customers are not all that sold on the existing incumbents" in the insurance industry, said Andrew Schwedel, a partner with Bain & Company. "That leaves them vulnerable to big tech as a category and somebody like Amazon in particular," because Amazon is known for convenience, he said.