NEW DELHI—An Apple Inc. AAPL -3.36% manufacturer has completed a trial run of the first-ever iPhones assembled in India, in an important step in the U.S. tech giant’s push into the fast-growing South Asian market.

The manufacturing of Apple’s cheapest iPhone model, the SE, was handled earlier this month by Taiwanese contract manufacturerWistron Corp. 3231 -1.43% , which has an assembling unit in the southern state of Karnataka, a state official with direct knowledge of the matter told The Wall Street Journal.

Apple said in a statement that it has begun initial production of a small number of iPhone SE handsets in Bangalore and will begin shipping the Indian-made devices to domestic customers this month. The first devices could hit stores as early as this week or next, according to a person familiar with the matter.

A Wistron spokeswoman said the company doesn’t comment on “market rumors or speculation.”

With sales cooling in China—long an engine for Apple’s growth—the Cupertino, Calif., company has been looking for new ways to build its brand in India. Apple has sought concessions on the taxes it pays to import some components, government officials say.

It is unclear how much the model might cost. Some Indian resellers already offer the iPhone SE for around $320, less than the $399 Apple charges in the U.S. Indian government officials hope the new price could be as much as $100 lower than the current $320. But Apple prefers to keep prices roughly consistent world-wide and aims to preserve its gross margins, a closely watched measure of profitability that reflects the percentage of revenue that remains after manufacturing costs.

Even though the SE, which has a 4-inch (10 cm) screen size, is Apple’s least expensive model, it is still out of reach of most Indians. Bringing its price down below $250 would help make it more affordable, analysts say, though it would still be well above the average smartphone price in India which research firm IDC says is around $150.

“Apple is likely to sell a good number of iPhones if it prices them so aggressively,” said Faisal Kawoosa, principal analyst at research firm CMR. “In three to five years, these users will be able to graduate to a standard-priced iPhone.”

Smartphone shipments in India grew 18% last year compared with just 3% globally, according to Counterpoint Research.

Apple Chief Executive Tim Cook has touted its sales in India, but Apple’s market share hasn’t risen above 3% since 2012, according to IDC.

Apple could seek more production in India in the future, the official said, adding that the state is eager to offer additional land for Apple contract manufacturers to expand operations.

Wistron Chairman Simon Lin and another executive, David Shen, are expected to meet senior politicians and officials of Karnataka toward the end of May to discuss the company’s broader plans, the official said.

Indian government officials hope Apple having a manufacturing base in the country could propel Prime Minister Narendra Modi’s Make in India push and create jobs.

“Apple coming to India is a [matter of] pride for us,” said R.V. Deshpande, Karnataka’s commerce and industries minister, referring to the SE production. “We are trying to get them in Karnataka as it’s the right place with all the required ecosystem.”

—Karan Deep Singh contributed to this article.