NEW DELHI—When top American naval engineers recently inspected India’s first locally made aircraft carrier they expected to find a near battle-ready ship set to help counter China’s growing sway in the Indian Ocean.

Instead, they discovered the carrier wouldn’t be operational for up to a decade and other shortcomings: no small missile system to defend itself, a limited ability to launch sorties and no defined strategy for how to use the ship in combat. The findings alarmed U.S. officials ​hoping to enlist India as a bulwark against China, people close to the meeting said.

“China’s navy will be the biggest in the world soon, and they’re definitely eyeing the Indian Ocean with ports planned in Sri Lanka and Bangladesh,” said retired Admiral Arun Prakash, the former commander of India’s navy. “The Indian navy is concerned about this.”

The February carrier inspection, in the port of Kochi, formed part of U.S. plans to share aircraft carrier technology with India. Indian naval officials followed up with a tour of an American shipbuilding yard in Virginia and strategy briefings at the Pentagon in September, the people close to the meetings said.

The U.S. and India are drawing closer politically and militarily. The two have participated in joint naval exercises with Japan. The U.S. has agreed to sell New Delhi everything from attack helicopters to artillery. Washington has approved proposals by Lockheed Martin and Boeing Co. to make advanced jet fighters in India. And in August, the two countries signed a military logistics-sharing accord.