Johny Srouji, the Apple executive in charge of building chips for the company’s devices, was fed up with Intel.

It was early 2017 and Apple was preparing a new line of iPhones to be released the next year, but the Intel modem for the devices, known as the 7560, wasn’t working properly, according to two people with knowledge of the relationship. The new iPhones would mark the first time Apple would rely exclusively on Intel for modems—a chip that handles the cellular connectivity in phones—instead of those from its longtime provider, Qualcomm. Intel had already overhauled the modem four times to bring it up to par with the latest Qualcomm modem. But missed deadlines and continuing technical issues with the chip were making Apple executives anxious, said one of the people.

“This would have never happened at Apple under my watch,” Mr. Srouji barked at his Intel counterpart, Venkata “Murthy” Renduchintala, during a meeting on Apple’s campus, according to the person, who was present at the meeting.