Prime Minister David Cameron has said Anglo-Irish relations are "at an all time high" as he met Taoiseach [Irish Prime Minister] Enda Kenny in London.
Mr Kenny is on a two-day visit to the UK and was invited to Downing Street for talks about the Northern Ireland peace process, the economy and trade.
He agreed that there were strengthening links between Ireland and the UK.
Mr Kenny said the Queen's Irish visit in 2011 had made an "extraordinary impact" on both countries.
As the two leaders held hour-long talks, Mr Cameron said: "I think we meet at a time when Anglo-Irish relations are at an all time high.
"I think relations between our countries are very, very strong but I still think there is even more we can do to strengthen our ties and strengthen our relationship."
The UK prime minister added that both states were returning to growth and that provided a "real opportunity" to improve economic links.
"In terms of trade, we have realised that one of the goals of the last summit we had when we talked about a joint trade mission between Britain, Northern Ireland and the Republic and we achieved that in Singapore, which I welcome."
Mr Kenny echoed Mr Cameron's remarks and said: "I do think that the visit of Her Majesty had an extraordinary impact on relations between the two countries.
"There's a building excitement in terms of the return visit from President Higgins."
Irish President Michael D Higgins is due to begin a three-day state visit to the UK in April.
He is the first Irish head of state to make a state visit to the UK, after accepting an invitation from the Queen.
The two leaders were also expected to discuss contentious issues in the Northern Ireland peace process.
They included the Haass proposals on dealing with flags, parade and the legacy of the Troubles, the 1989 murder of Belfast solicitor Pat Finucane and Army killings in Ballymurphy, west Belfast, in 1971.
The talks between Mr Cameron and Mr Kenny come two years after the two leaders released a joint statement pledging to work closer on issues such as the economy, Europe and Northern Ireland.
Previously the Irish government has called for a public inquiry into Pat Finucane's death.
They have also supported the formation of an independent panel to look at official documents relating to the deaths of 11 people, shot in 1971 by the Army in Ballymurphy.
Other topics due for discussion included a stalled wind energy deal between the Republic of Ireland and Britain.
The UK government had agreed to purchase substantial quantities of wind generated power but now appear to have cooled on the deal.
It is understood Mr Kenny is seeking to push through the agreement.
The pair are also reviewing visa arrangements that allow Asian visitors to travel between the Republic and the UK.
Mr Kenny said there was broad agreement on an initiative to have joint visas for visitors from India and China so that Ireland and the UK could be marketed as a single destination.
On Monday, the first day of his visit, the taoiseach was the guest of honour at a civic reception in Manchester, where he officially opened phase one of the new Irish World Heritage Centre.