The invitation was extended earlier in the day as the two held a meeting in Rome. Modi is currently on a visit to Italy to participate in the G20 summit.
"You have given the greatest gift, I am looking forward to visit India, Pope told ... Modi," the diplomat said at a press briefing in Rome.
The Indian prime minister described the meeting with the pontiff as a "very warm" one, adding they discussed a wide range of issues.
The last papal visit to India was made by John Paul II in 1999. In 2016, the sides were negotiating Pope Francis' visit to India the following year, but Indian Catholic Church leaders failed to convince Modi to invite him.