"I don't think he does understand the wider unionist culture of Northern Ireland", Foster said.
He said "my door is open" to the DUP after Ms Foster said Mr Barnier would not be hearing any pro-Brexit voices today.
"We need to agree rapidly by June the scope of alignment, what I call the safety controls that are.to respect the single market".
Three out of five Irish businesses are concerned at the potential impact of increased custom and certification procedures on trade between the Republic and Northern Ireland when the Uk leave the EU.
"The backstop we put in the draft treaty, is not there to change the United Kingdom red lines, it is there because of the United Kingdom red lines, because of the red lines on the customs union and single market", said Mr Barnier.
He added: "We want to succeed with the United Kingdom not against the UK".
"Together with the Irish Government we are looking for practical solutions".
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He then crosses the frontier to Newry in Co Down for meetings with business leaders. Mr Barnier will meet the Taoiseach Leo Varadkar and Tánaiste Simon Coveney.
Mr Barnier told reporters he wants Britain and the European Union to reach an agreement on the Irish border by June, a date he sees as a "stepping stone" towards October when the transition deal is due to be formalised.
"We have tried to get him to understand the unionist position for the people of Northern Ireland, but he hasn't really responded to that and I'm disappointed about that".
Barnier has proposed that Northern Ireland effectively stays in the single market and customs union following Brexit, a suggestion that has prompted claims that the EU is seeking to annex Northern Ireland.
In the meantime, the Irish Government is prioritising an agreement on the backstop or "Option C" which would ensure that in the event of there being no workable solutions forthcoming from the UK, Northern Ireland would remain within the EU's customs union, and would maintain regulatory alignment with the EU in order to avoid a hard border.
"The type of EU-UK free trade agreement that now seems likely would amount to a significant deterioration of the current economic relationship and would come with a heavy economic cost", Ibec chief executive Danny McCoy said.