According to a Foreign Office source, the letters were received by Pakistan through diplomatic channels on Tuesday, in which the sentiments from Pakistan "were also appreciated”, DAWN reported.
In their response, Modi and Jaishankar wrote about comprehensive and fresh talks between India and Pakistan. They said that India desires normal and cooperative relations with all neighbours, including Pakistan, adding that it has always preferred progress and peace of the people.
The Indian External Affairs Ministry in a statement said Modi wrote that "it is important to build an environment of trust, free of terror, violence and hostility". Minister Jaishankar also emphasised the need for an "atmosphere free from the shadow of terror and violence".
The letters come on the heels of the first face-to-face interaction between Imran and Modi on the sidelines of the 19th Shanghai Cooperation Organisation (SCO) summit in Bishkek. Qureshi had described it as a "courtesy" meeting.
Earlier this month, Prime Minister Imran Khan, in a letter to felicitate Modi on assuming office for a second term, had renewed Pakistan's offer to hold dialogue with India to resolve all contentious issues. He had also telephoned him for congratulating him for his victory.
Foreign Minister Shah Mehmood Qureshi had also written a letter to the newly-appointed Jaishankar and congratulated him on assuming office.