Mr Kenny and his wife, Fionnuala, are meeting the Pope in the Vatican today.
Pope Francis will receive them in the papal library, at the Apostolic Palace, beside St Peter’s Square.
Mr Kenny briefly met the Pope in April 2014 in the Vatican, at the canonisations of John XXIII and John Paul II, and invited the Pontiff to Ireland on that occasion as well.
The Pope’s visit to Ireland for 2018 is at the request of the Conference of Irish Bishops.
Mr Kenny is expected to convey to the Pope that “the normal state courtesies and support will be extended to him” when he comes to Ireland.
Mr Kenny is also being accompanied on the trip by Ireland’s highest-ranking civil servant, the Department of the Taoiseach’s secretary general, Martin Fraser, as well as by Mr Kenny’s private secretary and a press officer.
The delegation will first visit the Irish ambassador to the Holy See, Emma Madigan, and the Irish embassy, which was reopened in 2014, after its controversial closure in 2011 by the last Fine Gael-Labour government. The closure was blamed on cutbacks.
The delegation will then proceed to the Apostolic Palace, where Mr and Mrs Kenny will meet the Pope.
Mr Kenny will afterwards hold a private meeting with Cardinal Parolin, the Vatican’s secretary of state.
Both the meetings with Pope Francis and the secretary of state are expected to focus on the planning of the visit in 2018.
More than 1m people turned out in the Phoenix Park, Dublin, in 1979, to greet John Paul II, the last time a pontiff visited Ireland.
Church officials have confirmed that they intend for Pope Francis’s visit to be for two days, as part of the World Meeting of the Family in Dublin in August 2018.
It is unclear whether Pope Francis will also visit the North.
Mr Kenny and Pope Francis are also expected to discuss bilateral relations between the two states, developments in the European Union, and migration.