Tesla Motors Inc.’s vice president of communications plans to leave the company and join Aston Martin, the luxury sports car maker, where he will join former colleague Andy Palmer, who took over as chief executive recently, a person familiar with the matter said.

Mr. Sproule, who joined Tesla in March, had been Nissan Motor Co.’s top communicator. A respected industry veteran, Mr. Sproule’s addition to Tesla was considered a coup for the Silicon Valley startup, which had gone nearly two years without a director of the public...