Prime Minister Sir Keir Starmer met Taoiseach (Irish prime minister) Simon Harris at his country residence, Chequers, on Wednesday evening.
Mr Harris said that Sir Keir accepted his invitation to visit Dublin on 7 September.
Sir Keir said it was time to “move on to the next chapter, for stronger and deeper ties between both countries”.
Speaking after the meeting the taoiseach said both he and the PM “want to see our teams intensively working on areas of mutual concern and benefit and being truthful that hasn’t been happening in the way, manner or frequency which it would have in the past and which it must in the future”.
Both leaders also agreed to a new forum in which both governments would work on different issues.
Mr Harris was asked if the Irish government would revoke its interstate case that it has taken against the UK.
However he said, while it was too early to say, that both governments would be working “step by step” on that front.
RTÉ reported that the taoiseach presented Sir Keir with a Donegal GAA jersey.
Both leaders have talked of their desire to “reset” British and Irish relations.
They previously spoke by phone after the Labour leader’s 4 July election victory.
It was expected that they would address a number of matters which have caused friction between the governments in recent years.
This includes Brexit matters and the UK Legacy Act to address Troubles-related issues, which led to a legal challenge by the Irish government.
Speaking ahead of the visit, Mr Harris said the invitation extended by Sir Keir was “a strong signal of the value he attaches to the friendship and closeness that saw our two countries achieve so much together in the past”.
He added: “I am fully committed to a stronger, mutually respectful and ambitious partnership between both countries, and to getting down to work to make this happen.”
The taoiseach’s visit to the UK coincides with the official state opening of Parliament in London and the King’s Speech, during which around 35 separate pieces of legislation were announced.
They will meet again on Thursday at the European Political Community Summit in Oxfordshire.
On Wednesday, former Northern Ireland Secretary Shaun Woodward said parts of the relationship between the UK and Ireland had become frayed, partly because of policies.
“But also frayed because I think there’s not really been the relationship that there needs to be between prime minister and taoiseach,” he told BBC News NI’s Good Morning Ulster programme.
“At the core of the Good Friday Agreement you have the prime minister and the taoiseach as guarantors of that peace agreement and the policies and the processes that flow from that.
“You really do need an active engagement from prime minister and taoiseach to make that work.”
Earlier this month, Sir Keir travelled to Stormont to meet with the main Northern Ireland parties.
He was accompanied by the new Northern Ireland Secretary Hilary Benn and his chief of staff and former senior Stormont civil servant Sue Gray.