MEMPHIS, Tenn.—Fred Smith bristles at any hint that FedEx Corp. , the global delivery giant he built over four decades, could be disrupted by a player such as Amazon.com Inc.

“You can’t just overnight decide, ‘I’m going to pick up from every person in the world and every business in the world and be able to deliver it to every other,’ ” Mr. Smith said in a recent interview. “It’s fantastical.”

“Fantastical” is one of Mr. Smith’s favorite words to describe the possibility of being beaten by new rivals. He also used it last December, following a disappointing earnings report, to fend off analysts asking about Amazon’s push into the delivery business.

The next day, FedEx’s stock fell more than 12%. Dave Clark, Amazon’s senior vice president of world-wide operations, tweeted: “Ho!Ho!Ho! Have a Fantastical Holiday everyone!!!,” with a photo of an Amazon Prime branded jet model.

Mr. Smith brushed aside the quip. “What I understand about him, he’s a very hard-nosed guy, and he’s a smartass,” he said. “If he wants to take a shot, take a shot.”