Britain must soon take "major steps" towards avoiding a physical border with Ireland after Brexit, European Commission President Jean-Claude Juncker warned on a visit to Dublin on Thursday.
The Irish government was supported by Juncker, who said:"Ireland is not alone".
European Union leaders will express concern on Friday over a lack of progress in talks with London on how to avoid reimposing controls at the border between British-ruled Northern Ireland and the Irish Republic after Brexit.
European leaders meet in Brussels next week but a deal is not expected to be struck on the divorce's implications for the 300-mile border with its multiple crossing points until at least another European Council summit in the autumn.
On Thursday Mrs Clinton met Taoiseach Leo Varadkar at Government Buildings in Dublin to discuss gender equality.
Mr Varadkar told the meeting he still believes a solution to the backstop is "achievable" by the October council, but "at the same time we also need to be prepared for the possibility that won't happen".
A total of 63 per cent of those polled across the United Kingdom said they would still vote leave even when faced with the possibility that Scotland, Wales or Northern Ireland could quit the union.
"We want to make it clear again and again that Ireland is not alone". It is backed by 26 member states...
"I am strongly against any temptation to try to isolate Ireland".
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Britain and the European Union want to keep a free flow of people and goods over the border without returning to checkpoints, symbols of the three decades of violence in Northern Ireland that was largely ended by a 1998 peace deal. He will later receive an honorary doctorate from the National University of Ireland and attend an official dinner hosted by Mr Varadkar in Dublin Castle.
Later Mr Juncker addressed both houses of the Irish parliament, the Oireachtas.
But he said that no deal is as good as membership of the EU.
Jean Claude Juncker and Taoiseach Leo Varadkar.
The future of the Irish border after Brexit is one of the most vexed outstanding issues facing negotiators in Brussels.
The backstop solution has been agreed in very general terms in December, but there has been no consensus since on how to put it in place in practice by Brexit day in March 2019.
"History will record the important role President Juncker has played at a very hard time".
She added: "As the Brexit debate rages on, I continue to believe in the value of the European Union, and of a Europe that is whole, free and at peace".
The Prime Minister also faces further disruption in Parliament, with pro-EU Tories plotting to amend the forthcoming trade and customs bills.