The Taoiseach has asked Irish staff in the embassy in London to examine the logistics.
Tourism Minister Brendan Griffin confirmed to the Irish Examiner that moves are afoot to formally invite the popular couple. William’s grandmother, Queen Elizabeth, visited here in 2011.
“This would be the next step and, of course, the next generation in Anglo-Irish relations, with the young couple visiting.”
Mr Varadkar spoke to Dan Mulhall, the Irish ambassador to London, about the potential visit when he was there to meet British prime minister, Theresa May.
While nothing has been confirmed from London or Dublin, Mr Varadkar and the Government want to ensure that any formal invite would only be issued once there was an an acceptance of an Irish welcome for the couple.
The timing of any trip comes at a difficult period for Britain, as it negotiates its exit from the European Union.
The Duke and Duchess of Cambridge have already made a number of trips, partly organised by the Foreign Office in Britain, to EU capitals, including France, Germany, and Poland this year.
The visits have prompted suggestions the royal couple are being seen as unofficial Brexit ambassadors, in moves designed to charm leaders.
A diplomatic challenge would be any attempt to match the success of Queen Elizabeth’s visit here in mid-2011, which both British and Irish governments agreed helped Anglo-Irish relations.
Officials in the department of sports and tourism also strongly believe that a visit by Prince William and Kate Middleton would again strongly boost the image of Ireland as a holiday destination for British travellers, especially at a time when numbers coming here have fallen on the back of Brexit.
Prince William has publicly described, in media interviews, Queen Elizabeth II’s delight and happiness at her successful visit to Ireland in 2011.
The prince, second in line to the British throne after his father, Prince Charles, also led a ceremony with former taoiseach, Enda Kenny, in Belgium last month, marking the 100th anniversary of the Battle of Messines, in World War I.