Such a measure would mean that the UK could impose customs checks between Great Britain and Northern Ireland to solve the issue of a border with Ireland - which will remain an EU member while the neighbouring UK territory Northern Ireland would not after March 2019.
Barnier told reporters in Brussels that he did not believe such checks would amount to a border across British territory, adding: "Somewhere in airports and ports there would be controls, but I would not refer to a border".
The plan involves keeping Northern Ireland in the EU Customs Union and Single Market, effectively subject to the rules applicable to all member-countries.
He claimed that the EU was offering this as a "backstop option", which is not intended to "provoke" the UK but to encourage it to come up with workable alternatives.
However, May shot the option down during Prime Minister's Questions session in the House of Commons soon after.
"The draft legal text the commission has published would, if implemented, undermine the UK common market and threaten the constitutional integrity of the UK by creating a customs and regulatory border down the Irish Sea, and no UK Prime Minster could ever agree to it," she said.
There are fears that Brexit
could lead to the return of border checks between Ireland and Northern Ireland, which would threaten cross-border trade. Such checks are, according to the EU, needed because Britain will no longer be subject to the bloc's rules on, for example, animal health and other standards and Brexit will create a land border with the EU.
The Irish government said that the option offered in the draft was "very much a default and would only apply should it prove necessary".
Ireland's Indian-origin Prime Minister Leo Varadkar hit back at hard Brexiteers like UK Foreign Secretary Boris Johnson for failing to come up with any workable solution to the border issue.
"It's really up to those who can't accept the backstop to actually come up with those alternative solutions. It can't be theoretical stuff," he said.
Johnson reacted by accusing the EU of politically using the issue of the Northern Irish border "to try and keep the UK in the customs union effectively the single market so we cannot really leave the EU".
Britain's Opposition Labour party said it would solve the Irish border question by entering into a new customs union with the EU, meaning checks are not needed as people and goods pass between Northern Ireland and the Republic of Ireland.
The timing of the EU's draft text is being seen as the 28-nation economic bloc's attempt to put pressure on May to spell out the exact nature of a future relationship between Britain and the EU as she gets set to deliver a key Brexit policy speech on Friday.
Donald Tusk, the president of the European Council, is set to travel to London tomorrow to pile yet more pressure on May to explain in her speech how she plans to solve the numerous Brexit challenges.
There is also opposition in Britain to any role for the European Court of Justice after Brexit, while the EU is proposing that disputes over the Brexit agreement in future years be settled by a "joint committee" which can refer to the EU's court for a binding decision.
Meanwhile, former British prime minister John Major made a significant intervention with a speech today, calling for a second referendum to allow the public to change its mind.
Major, who led the Conservative Party between 1990 and 1997, said May should offer British MPs a free vote on the final deal, with the option of putting it to the public in a second referendum.
"Many electors know they were misled. Many more are beginning to realise it. So, the electorate has every right to reconsider their decision," he said.
Last week, May had finally succeeded in uniting the Remainers and Leavers in her own Brexit war Cabinet to some degree after a gruelling session of discussions and presentations.
Her much-anticipated policy speech on Friday is expected to lay out the tone for trade negotiations with the EU to begin in April.
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