ADIBATLA, India—Having signed a string of multibillion-dollar orders from foreign firms to make parts for helicopters, aircraft and trains over the past couple of years, India is struggling to find people with the skills to build them.

In a $3.3 billion push, India is racing to equip 15 million people by 2020 with the skills necessary to realize Indian Prime Minister Narendra Modi’s aim to bring more high-grade manufacturing to the country.

The challenges are significant as foreign suppliers including Boeing Co., Airbus Group SE and Alstom SA often can’t find the employees with the training and experience to help fulfill Mr. Modi’s “Make in India” program.

More than 80% of engineers in India are “unemployable,” Aspiring Minds, an Indian employability assessment firm, said in a January report after a study of about 150,000 engineering students in around 650 engineering colleges in the country.

A lack of specialized courses mean companies have to train their own people from scratch.