NEW DELHI—Apple Inc. will soon start assembling iPhones in India for the first time, say government officials familiar with its plans, boosting the company’s chances of gaining a foothold in the fast-growing market.

Taiwanese contract manufacturer Wistron Corp. will likely start making iPhone 6 and 6S models here in the next four-to-six weeks at its plant in Bangalore, said an official of the southern state of Karnataka where the tech hub is located. It will add Apple’s cheapest iPhone model, the SE, to its assembly line in...