Prince Charles has visited the Irish Embassy in London, making him the first member of the British Royal Family to do so.
The prince, joined by the Duchess of Cornwall, met leading figures from Irish society at a special reception.
Broadcaster Sir Terry Wogan described the visit as "significant".
The Queen has not set foot in the Republic of Ireland during her 57-year reign but is expected to make her first visit next year.
No British monarch has visited the country since it gained independence early in the last century, but other Royals have.
There has been much speculation about a possible trip, but Buckingham Palace has said it does not talk about the Queen's diary so far in advance and there has been no official invitation.
Sir Terry said the Queen would be welcomed in the Irish Republic.
"I've been in Ireland a lot this year doing a documentary for BBC television on Ireland and there is a lot of anticipation," he said.
"You've got to understand despite 700 years of oppression, starvation and immigration, the Irish still have an enormous affection for the English.
"This is an historic thing, the Queen will undoubtedly get a fantastic reception in Ireland."
The embassy event was hosted by the country's ambassador, Bobby McDonagh, and guests included Bob Geldof, former Formula One racing boss Eddie Jordan, presenter Eamonn Holmes and comic Patrick Kielty.
Speaking about the possibility of a Royal visit, Bob Geldof said it would be important for both Britain and Ireland: "The time is right now, it's a sign of maturity and getting a life for both of us."
Northern Ireland MPs and peers were also at the event in London's Belgravia.
The Reverend Ian Paisley, now Lord Bannside, also had a positive response to the rumoured visit, saying: "It's very important that the relationship between Ireland and the UK should be developed.
"We are part of the EU together and have many things in common and in a period of difficulty we should be co-operating with one another."
The guests were treated to Irish black pudding and cheeses, accompanied by half pints of Guinness.
George V was the last British monarch to visit Ireland in 1911 but, for much of the time since then, Anglo-Irish relations have been fraught.
The bitterness caused by the partition of Ireland a decade later and the use of the British army in the Troubles in Northern Ireland, strained relations between the UK and the Republic of Ireland for much of the 20th Century.
But the success of the peace process in Northern Ireland has greatly eased tensions between the two nations.