The Korea JoongAng Daily newspaper, quoting a senior KIA official, said the deal is worth about $200 million.
Four trainers will be completely manufactured in South Korea. The remaining 16 will be assembled in Peru by KAI and Seman Peru, a state-run company under the control of Peru's Ministry of Defense.
No timeline for delivery of the aircraft was given.
The KT-1 is a single-engine, two-seat trainer with a maximum speed of about 356 miles per hour. It was first produced by South Korea in 1998 and has been exported to Indonesia and Turkey.
The deal follows a meeting in June between South Korean President Lee Myung-bak and Peruvian President Ollanta Humala at a U.N. event and a visit to Seoul by Humala the month before.
"The Latin American trainer market is dominated by Brazil and Switzerland," said KAI Vice President Park Noh-sun. "In particular, it's significant that Korea won the deal after competing against Brazil's Embraer S.A., which controls 40 percent of Latin America's mid-size passenger plane market."
The report quoted Oh Young-ho, president of the Korea Trade-Investment Promotion Agency, as saying that the joint-manufacture provision of the contract helped clinch the deal with Peru.