Moon, who arrived here on Sunday on his first state visit to India, was accorded a ceremonial welcome at the Rashtrapati Bhavan by President Ram Nath Kovind and the prime minister this morning.
The South Korean president also paid floral tributes at the memorial of Mahatma Gandhi at Rajghat.
During the talks between Modi and Moon, key issues such as the situation in the Korean peninsula, and ways to boost trade and defence cooperation, were expected to be discussed.
Yesterday, Moon had jointly inaugurated the world's largest mobile phone factory in Noida with Prime Minister Modi. The two leaders had also travelled in the Metro to Noida while heading for the inauguration of the Samsung facility.
Moon was treated to a colourful traditional Indian dance performance here last night that was organised especially for him by Prime Minister Modi.
President Moon and his wife Kim Jung-sook hosted a dinner last night for the Korean community living in India, and greeted a group of Indian veterans who had assisted the South during the Korean war.
External Affairs Minister Sushma Swaraj had called on President Moon yesterday and discussed ways to elevate the Special Strategic Partnership between the two countries.