Following the meeting, Mr Barnier said: "The EU is looking for practical solutions to avoiding a hard border, in full respect of the constitutional status of Northern Ireland, as set out in the Good Friday Agreement".
While May dismissed the European Union text as unacceptable, Chancellor of the Exchequer Philip Hammond told lawmakers on Monday there were no plans to present an alternative draft treaty before a summit later this month.
"That is his interpretation, we don't think it's a fair interpretation of the joint report from December and therefore work needs to be done on this now".
Northern Ireland will be the UK's only border with the EU.
Barnier conceded, however, in a press conference last week that there may need to be checks on goods coming from Northern Ireland at ports and airports in the rest of the United Kingdom should the proposal for "full alignment" be implemented.
The Sinn Féin leader added that there could be no overall agreement on Brexit unless the issue of the Irish border was solved.
She said companies needed to have a plan in place to deal with the potential fall-out, especially those exposed to the British market.
He added: "It is driving a coach and horses through the EU's own principles about no hard borders". The only way to avoid this post-Brexit is for regulations on both sides to remain more or less the same in key areas including food, animal welfare, medicines and product safety.
Normalcy returns to Kashmir Valley after protests, shutdown over Shopian firing
Examinations for various classes by J&K School Board and Kashmir University have also been postponed. He has been identified as Gowhar Ahmed Lone of Molu-Chitragam village of Shopian .
The research shows a high level of ongoing product turnover among trading ﬁrms who regularly introduce new products to the market, with many goods traded for one year only.
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And Nigel Dodds, Mrs Foster's deputy and the DUP's Westminster leader, also blasted Mr Barnier's proposals. Placing a border down the Irish Sea would not just be politically unacceptable but would be economically catastrophic.
"There are three options in relation to future relations, this [legal text] has only reflected the third of those options, it doesn't reflect the first and second".
Those proposals should not be flippantly dismissed, Foster said.
"We feel that the current draft legal text has omissions in it".
"What emerges clearly from this research is the high degree of integration of the economies north and south". The European Union's skewed interpretation of "Option C", paragraph 49 of the phase 1 agreement represented one of the most audacious examples of EU encroachment and interference - the very thing many in the United Kingdom disliked and rejected during the EU referendum.
Foster appeared to offer support for the prime minister's suggestions that lessons could be learned from the Canada-US border.
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