Apple Inc.’s new iPhone, which is expected to be unveiled Tuesday, was plagued by production glitches this summer, according to people familiar with the situation, which could result in extended supply shortfalls when customers start ordering the device later this month.

New iPhones typically are in short supply when first released. But if shortages of the new phone extend beyond the initial sales period, which is expected to begin Sept. 22, analysts and investors could dial back their projections for sales in the crucial...