Tarias and Foreign Minister Simon Coveney said it is possible to conclude a Brexit agreement this month, and a number of advances have been made in recent weeks.
Mr Coveney said that he and Justice Minister Charlie Flanagan welcomed Cabinet Secretary of State David Lidington and Karen Bradley, Secretary of State of Northern Ireland, at the Ivoagh House British-Irish Intergovernmental Conference (BIIGC).
Both Mr Coveney and Mr Lidington have said they are very close to resolving the differences.
Mr Coveney said the negotiating teams worked hard to provide a way to curb British fears on the backstop, adding that Dublin and the EU can not be flexible in the "all weather" disadvantages.
He said that there is a need for movement on the UK side to find the legal wording of the termination contract.
He said there is a need to ensure that the transaction is politically supported.
Mr Lidington said that the British government was committed to getting the ultimate difficulties of negotiations as swiftly as possible.
He said he had been a positive movement over the last few weeks and said that the UK government would have to face its commitments on the border issue.
He added that it is more difficult to continue the power distribution in Northern Ireland, while the question of Brexit is still unresolved.
He said he hopes and expects a final agreement in the coming weeks.
Arlene Foster, head of the DUP, says the United Kingdom and the EU are close to a Brexit case that will work in Northern Ireland.
"Goodness, we are here several times and I think we are close to a contract that will work in Northern Ireland, that's what we want," he told reporters after meeting British Prime Minister Brexit, Dominic Raab.
Mrs Foster said that he repeated Mr Raab's opposition to the opposition against further checks between Northern Ireland and the rest of the United Kingdom.
His party delegation, including Northern Belfast MP Nigel Dodds, MEP Diane Dodds, and South Belfast MLA Christopher Stalford met this afternoon with the Stormont House, Mr. Raab.
Mrs Foster said that his party made it clear to Mr Raab that "from a constitutional point of view, but from an economic point of view, it is very important that no regulatory barriers exist in the absence of tariff barriers".
"I'm glad you took the opportunity to visit the ports today because you saw what was happening in the ports," he said.
"We are continuing in an ongoing process, we want to be able to assist in the negotiations, we want to reach an agreement to exit the European Union, which is good for Northern Ireland and only for Northern Ireland if it is to remain part of the UK."
Meanwhile, President Sinn Fein, Mary-Lou McDonald, said he had a direct meeting with Mr Raab when he warned him not to "go without thinking".
Party leader in Northern Ireland Michelle O & # 39; Neill was also in the delegation where he met with Raab.
"There is a direct meeting we call Dominic Raab, and we have once more clearly formulated the need to prevent the fundamental economic, social and political prosperity of the island of Ireland," McDonald said.
The meeting of major political parties in Bradley and Northern Ireland in Belfast yesterday to discuss the restoration of formal power distribution, SDLP leader Colum Eastwood's "waste of time".
The four members of the five parties expressed frustration after the meeting and urged Mrs Bradley to convene multiparty talks.
The BIIGC established in 1998 under the Great Friday Agreement focuses on a bilateral agreement between the two countries.
The lack of the Brexit negotiations and the division of power in Northern Ireland is at the heart of the conference agenda.
North-South Security Cooperation and bilateral cooperation between the British and Irish governments.
The latest BIIGC meeting took place in July.
Earlier this week Mrs Bradley told the House of Deputies to "actively consider" how and when an external mediator could play a "constructive" role in re-negotiating the negotiations.
He told MEPs that he would raise the Irish government at this conference.