Fears of a return to violence have been raised if the Good Friday Agreement is damaged by the imposition of a hard border on the island of Ireland after Brexit.
Brussels had suggested a solution to the problem – a common regulatory area for goods and customs with the rest of the EU.
But this has so far been rejected by the UK as “annexing” Northern Ireland.
However in a major breakthrough for Britain, Mr Barnier described the issue as “the biggest risk” caused by Brexit and said he was “ready to improve” the EU’s proposal.
He said: “Since we will not know what the future relationship will bring by autumn 2018, we need to have a backstop solution in the withdrawal agreement.
“The UK agrees with this, and both the EU and the UK have said that a better solution in the future relationship could replace the backstop.
“What the EU has proposed is that Northern Ireland remains in a common regulatory area for goods and customs with the rest of the EU.
“We are ready to improve the text of our proposal with the UK.”
Michel Barnier was also positive about reaching a deal “unprecedented in scope and depth” around free trade.
He warned that UK proposals must not “undermine” the four pillars of the single market – free movement of goods, capital, services and labour, which he called “one of the EU’s biggest achievements” – by seeking freedom only for goods.
“But I remain confident that the negotiations can reach a good outcome.”
“It is possible to respect EU principles and create a new and ambitious partnership.”
Mr Barnier has previously said he does not want to “dramatise” the backstop solutions.
The EU’s negotiator said: “Let me be clear – we are not asking for any border between Northern Ireland and any other part of the UK.
“We must de-dramatise the backstop.”
The thorny issues of the Irish backstop has been at the heart of the Brexit negotiations.
Prime Minister Theresa May would need would prevent a hard border between Northern Ireland and the Republic of Ireland.
But the UK is opposed to a backstop, which would create a customs border in the Irish Sea.
It comes after Sinn Fein leader Mary Lou McDonald leader declared a referendum to decide whether Northern Ireland should split from the UK before Brexit must be held “as soon as possible”,
In a shock U-turn on the party’s policy on an independence, she reversed her position and insisted a vote should be held sooner rather than later – despite the ongoing Brexit talks.
Mrs McDonald acknowledged a vote during the negotiations “does not provide the best environment for a referendum on Irish Unity”.
But she added “Sinn Fein wants to see a referendum as soon as possible”, the Belfast Telegraph reports.