Amazon Chief Executive Jeff Bezos is set to finalize the purchase of three New York apartments Tuesday in a deal valued at around $80 million, according to people familiar with the transaction.

Mr. Bezos is buying a penthouse and two units directly beneath it at 212 Fifth Avenue, located near Madison Square Park, these people said. The main penthouse alone spans three floors, with a private elevator and large terraces, according to StreetEasy. If turned into a single-family home the three units would total more than 17,000...