Earlier today, Barnier stated that the EU has advanced on "some separation issues" for which European businesses need certainty - such as customs, Value-Added Tax and certificates for goods.
The taoiseach did not make the summit dinner last night because had business in Dublin with European Commission President Jean-Claude Juncker.
Officials in Dublin and Brussels had initially set the European Council summit next week as an important deadline for progress on Northern Ireland, but all sides now accept that EU leaders at the meeting will not be able to herald a significant breakthrough.
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.
The call comes ahead of a meeting of the British-Irish Council (BIC) in Guernsey on Friday, which will bring the leaders together with Mrs May's deputy, David Lidington, and Irish Prime Minister Leo Varadkar.
Mr Juncker will be accompanied on his engagement in Dublin on Thursday morning by Mr Barnier and European Union agriculture commissioner Phil Hogan.
The future of the Irish border after Brexit is one of the most vexed outstanding issues facing negotiators in Brussels. It is backed by 26 member states...
Juncker said he was "strongly against any temptation" to try to isolate Ireland and not to conclude a withdrawal deal.More news: Greece gets 'historic' debt relief deal
Addressing both houses of the Irish Parliament in Dublin, he said time was running out to find a solution to the border issue.
The visit comes as European Union member nations are being warned to step up preparations for a no-deal Brexit.
Mr Juncker met Mr Higgins and his wife Sabina and there were warm greetings as he thanked his hosts.
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.
Another paragraph clearly states the temporary customs arrangement will be "time-limited" and will come into force after the Brexit transition period ends in December 2020.
"We need to see much more progress from the United Kingdom to implement the commitments they made in December and March, and I expect the UK's efforts to intensify in the period ahead".
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".
"We will also discuss other key items on the agenda of next week's meeting of the European Council such as trade, ensuring a stable euro, and equipping the Union with a budget for its future needs".