The EU has said that it wants to find "flexible and imaginative solutions" to the issue of a post-Brexit border on the island of Ireland.
His comments come as the European Commission published a new position paper, stressing that the Good Friday Agreement continue to be protected "and strengthened in all its parts" after Brexit.
It says nothing can affect the country's membership even though the United Kingdom government has already said it would leave the customs union.
The European Union is publishing series of papers this week highlighting its Brexit position and its approach to existing issues including the Irish border. "The U.K. wants to use Ireland as a kind of test-case for the future EU-U.K. customs relations - this will not happen".
They pinned the blame on Britain's failure to make the necessary "sufficient progress" on a financial settlement for Brexit, on citizens' rights and on the border in Ireland.
According to an European Union document on the issue of Ireland's relationship with the United Kingdom after Brexit, leaked to the Guardian on Wednesday, Brussels believes that the United Kingdom bears the "onus to present solutions which overcome the challenges" created for the region by the Brexit vote.
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In its position paper last month, London said the issue of how goods and people will move across the Irish frontier can not be separated from discussion about wider customs arrangements between Britain and the European Union. Many EU officials are concerned the U.K.'s own policy document implied the Irish border was Europe's responsibility after Britain's withdrawal from the bloc.
"All this could be avoided if the British government took the sensible decision, on leaving the EU, to remain in the single market or at least the customs union".
Barnier met First Minister Nicola Sturgeon in July of this year, despite Scotland and Wales being denied any input into Brexit negotiations by the UK Government.
In a further development, the European Parliament's Conference of Presidents met on September 7, where concerns were raised as to the lack of progress made in negotiations.
A spokesperson said: "The UK Government, along with the Irish government, is engaging intensively with the Northern Ireland parties to secure the reestablishment of inclusive, devolved government at Stormont, and the operation of all the institutions established under the Belfast Agreement".
Committee deputy convener, Lewis Macdonald, said the Brexit talks were " historic negotiations that will have a profound impact on Scotland" as he welcomed Mr Barnier's "willingness to engage with the committee".