Inside a room of the ornately decorated Hotel du Palais during last month’s Group of Seven summit in Biarritz, France, President Trump awaited a meeting with Egyptian President Abdel Fattah Al Sisi.

“Where’s my favorite dictator?” Mr. Trump called out in a voice loud enough to be heard by the small gathering of American and Egyptian officials. Several people who were in the room at the time said they heard the question.

The...