BANGALORE, India—Call-center company 24/7 Customer Pvt. Ltd. is desperate to find new recruits who can answer questions by phone and email. It wants to hire 3,000 people this year. Yet in this country of 1.2 billion people, that is beginning to look like an impossible goal.

So few of the high school and college graduates who come through the door can communicate effectively in English, and so many lack a grasp of educational basics such as reading comprehension, that the company can hire just three out of every 100 applicants.

India projects an image of a nation churning out hundreds of thousands of students every year who are well educated, a looming threat to the better-paid middle-class workers of the West. Their abilities in math have been cited by President Barack Obama as a reason why the U.S. is facing competitive challenges.

Yet 24/7 Customer's experience tells a very different story. Its increasing difficulty finding competent employees in India has forced the company to expand its search to the Philippines and Nicaragua. Most of its 8,000 employees are now based outside of India.

In the nation that made offshoring a household word, 24/7 finds itself so short of talent that it is having to offshore.