Amazon.com Inc. plans to open a store in the middle of New York City, according to people familiar with the plans, the first brick-and-mortar outlet in its 20-year history and an experiment to provide the type of face-to-face experience found at traditional retailers.

The site, set to open in time for the holiday-shopping season on the same busy street as Macy’s Inc.’s flagship store, would mark an attempt by Amazon to connect with customers in the physical world. Amazon has built its business on competitive pricing and fast...