Yesterday, President of the European Commission Jean-Claude Juncker said Ireland would come first in negotiations, as he told a joint sitting of the Dáil and Seanad that Ireland's border was Europe's border and it remained a priority.
Mr Juncker started his visit with a meeting with Mr Varadkar and Mr Coveney at Government Buildings.
'I am strongly against any temptation to isolate Ireland and not to conclude the deal on Ireland.
He argued the uncertainty as to whether people and goods will still be able to travel freely across the 300 mile Northern Ireland/Ireland border after Brexit was feeling the nationalist sentiment.
While both sides agree they must have a contingency plan to keep the border open, a so-called backstop solution was only agreed in very general terms in December and there is no consensus on how to put it in place by Brexit day in March 2019.
Mr Juncker has raised the prospect of Britain leaving without a deal during his visit to Dublin.
The council will also talk about the political situation in Northern Ireland and Mr Varadkar is expected to reiterate his regret that no Northern Ireland ministers will be at the British-Irish Council summit.
"It's a point that has also been reinforced by Michel Barnier, the European Commission's chief negotiator for Article 50 negotiations", said McGuinness.
"There is now an urgent need to intensify our efforts " to achieve a withdrawal agreement between the European Union and the U.K., Irish Prime Minister Leo Varadkar has warned Thursday.
We must ‘immediately’ return undocumented immigrants ‘with no judges or court cases’
He said , "When somebody comes in, we must immediately, with no Judges or Court Cases, bring them back from where they came". Immigrant rights activists have condemned the proposal, warning that it could violate due process requirements.
Advances have been made in a number of areas, such as customs, Value-Added Tax and nuclear waste regulation, European Union chief negotiator Michel Barnier has said, but "a lot more work" needs to be done to meet the October deadline.
On Friday President Juncker met Irish president Michael D Higgins at his official residence in Dublin's Phoenix Park.
Earlier this year, Mrs May rejected the EU's own proposed backstop solution to the Irish border issue.
Both the European Union and United Kingdom have agreed to include a so-called "backstop" option in the withdrawal treaty, which would commit the United Kingdom to align with an European Union regulatory framework in the absence of a wider trade deal.
Secretary of State Karen Bradley and Irish Prime Minister (Taoiseach) Leo Varadkar will both be in attendance.
Britain's plan for a temporary customs "backstop" is "not acceptable" and any fall-back option can not be time-limited, the European Parliament's chief Brexit co-ordinator has told MPs.
The Prime Minister also faces further disruption in Parliament, with pro-EU Tories plotting to amend the forthcoming trade and customs bills.
The marine environment is listed as the main topic for discussion, but Brexit is expected to be debated ahead of next week's European Union summit in Brussels.