Overall, 32 per cent agreed that it was more important to avoid creating a hard border, "even if this meant the United Kingdom not being able to do its own trade deals with non-EU countries", while 41 per cent said the United Kingdom should leave the customs union "even if this means a hard border with customs checks".
Coming a week ahead of a crunch European Council meeting of leaders in Brussels, the visit is a clear demonstration of the EU's solidarity with Ireland's position.
"We have advanced on some separation issues for which European businesses need certainty, such as customs, VAT, Euratom and certificates for goods", Barnier said in a statement accompanying a joint declaration from Brussels and London.
The Taoiseach said: "I'm very much looking forward to President Juncker's first visit to Ireland as Commission President".
The BIC brings together representatives of the Irish and British governments, the devolved administrations in Northern Ireland, Scotland and Wales and the Isle of Man, Guernsey and Jersey crown dependencies.
Lord Adonis, who supports the Best for Britain anti-Brexit campaign, said: "Every day that the Prime Minister spends arguing with her own Cabinet and manoeuvring to rewrite the British constitution is a wasted day when it comes to our negotiations with our European Union partners".
The European Commission president addressed the Oireachtas this afternoon, and received applause when he said the Irish border is also Europe's border.
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"The UK is clear that the future customs arrangement needs to deliver on the commitments made in relation to Northern Ireland. It remains our intention to achieve a close economic partnership that does not require the backstop to be in operation".
He was applauded as he told members of the Dail and Seanad that "Ireland will come first" in the Brexit negotiations.
A time-limited "backstop" option agreed in principle but not in detail between Britain and the European Union has been interpreted by Europe as meaning Northern Ireland would effectively stay within the bloc's regulatory rules on issues like customs to preserve the now invisible and frictionless border arrangements.
Mr Juncker will conclude his two-day visit to Dublin on Friday when he meets President Michael D Higgins at his official residence and then tours the Croke Park Gaelic Athletic Association (GAA) stadium.
"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".
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.
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.