NEW DELHI—President-elect Donald Trump doesn’t take office in Washington until Friday, but he is already forcing firms in India’s mammoth $108 billion technology-outsourcing industry to rethink their hiring practices in the U.S., their largest market.

While Mr. Trump has chastised U.S. firms for offshoring American jobs, Indian outsourcing firms could be set to see renewed heat for doing the opposite—placing foreign workers in the U.S., mainly through a skilled-worker visa, known as the H-1B. Faced with the prospect of possible...