The strength in opinion carried across political divides.
Mr Juncker said the Republic "will come first" as Europe counts down to the UK's departure from the EU.
I do not doubt the sincerity of the European Union negotiating team and their commitment to Ireland in these negotiations but serious issues still remain unresolved with President Junker himself acknowledging that "there is not much time left to reach agreement".
After the meeting, Mr Juncker made clear that a resolution to the border issue was crucial to the withdrawal treaty.
Across Great Britain, 62 per cent of respondents said Brexit negotiations were proceeding too slowly, with 30 per cent blaming the delays on "politicians in Britain who want a soft Brexit or to stop Brexit altogether".
"If Mr Juncker was really focused on helping Ireland, he would be encouraging the European Union to focus on getting a post-Brexit trade deal in place".
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 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.
"I am strongly against any temptation to try and isolate Ireland and not to conclude a deal on Ireland".
The European Commission President was handed a Cork shirt with the number 27 in acknowledgement of his role representing the EU27 members.
Ms Sturgeon and Mr Jones said the European Union had been "very clear" that the UK Government's red lines meant the "only Brexit on offer is one which will deeply damage our economies and possibly jeopardise our security".
"I will present the state of play of the negotiations to the European Council and the European Parliament".
Leaders of the 27 countries that will remain in the European Union after Britain leaves next year will take stock of negotiations on a withdrawal agreement and on a political declaration on what future relations with London should look like.
Mr Junker will receive an honorary doctorate from the National University of Ireland and attend an official dinner hosted by Mr Varadkar in Dublin Castle later on Thursday.
At the start of Mr Juncker's address, he joked that he wished he was drunk.
"Our focus is on getting the best deal possible for the United Kingdom - one that allows us to take back control of our borders, laws and money". The U.K. wants the backstop to be time-limited but Varadkar said it could not have an "expiry date".
On Friday he will meet President Michael D Higgins and visit Croke Park where he will see a demonstration of gaelic football and hurling.