EU leaders say a deal on the land border between Ireland and Northern Ireland, which stayed in the United Kingdom after Irish independence in 1922 and suffered decades of deadly violence before the arrival of the European single market and a peace accord, is on the priority list as talks loom.
Barnier addressed Irish lawmakers on Thursday, telling them he would work "to avoid a hard border" but stressing that Britain's decision to leave meant there would be "consequences" such as customs checks.
He reassured them he would do his best to avoid a hard border between Ireland and Northern Ireland post-Brexit and aim for a deal that respects the Good Friday Agreement in "all of its dimensions".
Barnier suggested that Brussels would be open to considering "flexible and creative" solutions to the customs issue, but said these would have to "respect the integrity of the European Union legal order". Wicklow, Mr Blair told reporters that Northern Ireland should be designated as a "special case".
Northern Ireland voted to remain in the European Union during last summer's Brexit referendum.
And that will take a couple of years at least, assuming the Brexit timetable is maintained The border doesn't exist in an Irish bubble.
However, the Irish Taoiseach (prime minister) disagrees, saying he believes now is not the time for a vote in his address to Barnier.
"We are in this negotiation together and a united European Union will be here for you", he said.
The issue of the border between the Republic of Ireland and its neighbours Northern Ireland has raised its head once again, with many pointing out that an Ireland remaining in the bloc could provide back-door access to the UK.
But according to a person who was present, Mr Blair said that the weak leadership of the British Labour Party by Jeremy Corbyn had facilitated Brexit, by having nobody to put a strong counter argument.
Over the Past 150 years or so, it was frequently said that as soon as Britain found an answer to the Irish question, the Irish changed the question.
BBC Northern Ireland's economics and business editor John Campbell said the free movement of goods was likely to be a key area of negotiations - and agricultural food companies in particular were extremely concerned about the potential for tariffs which could wreck their business.
"But it really would be a disaster to have a hard border".