Speaking briefly to the press outside before engaging in a private meeting he said "I am very happy to be in Newry, coming back to Northern Ireland in the context of this very very hard complex and extraordinary negotiations of Brexit".
Backstop refers to a proposed agreement between the European Union and Britain to ensure the border between Ireland and Northern Ireland will remain the same as it is now after Britian's withdrawal from the EU.
She suggested that the EU's chief negotiator did not "respect" the unionist position for the future of Northern Ireland and was not acting as an "honest broker" during talks.
"Michel Barnier's trying to present himself as someone who cares deeply about Northern Ireland, and if that is the case he needs to hear the fact that we are part of the United Kingdom (and) will remain part of the United Kingdom constitutionally, politically, and economically", Foster told the BBC.
The Northern Ireland-Ireland border will be the U.K.'s only land frontier with the European Union after Brexit.
"Therefore his proposal of us being in an all-Ireland regulatory scenario with a border down the Irish Sea simply does not work".
Michel Barnier, Chief EU Brexit negotiator speaking to the press at InterTradeIreland in Newry.More news: Senate approves Pompeo as US Secretary of State
She said: "The Taoiseach and Irish Foreign Affairs Minister have been very willing to involve themselves in issues affecting border communities in Northern Ireland, not least in relation to concerns surrounding the UK's exit from the EU.
Mrs Foster and some others need to understand that and respect that", Barnier told a press conference with Irish Prime Minister Leo Varadkar.
Mr Barnier said: "We have no intention of questioning the UK's constitutional order".
"We want to see all parts of the Agreement operating and I also want to see the great strides that we have made on North/South cooperation continue and grow in the years ahead".
He added: "The backstop is not there to change the UK's red lines".
He inadvertently had a dig at Davis when he took a question from a Northern Irish journalist about Davis believing the backstop wasn't necessary. These days, in contrast, one can drive across the border without noticing it. Security measures were phased out following the Good Friday peace agreement of 1998, and the EU's single market and customs union have banished the need for inspections of imports and exports between the two countries.If the United Kingdom leaves the single market and customs union, as Theresa May's government intends, those inspections will return in some form. "Together with the Irish government we are looking for practical solutions".