The European Union's chief Brexit negotiator, Michel Barnier, has met with Irish MEPs in Strasbourg.
His meeting followed British Prime Minister Theresa May's address on Brexit in London earlier.
In a speech outlining her government's plans, Mrs May said both British Houses of Parliament will vote on the final deal.
Mrs May also said the plans "cannot mean membership of the single market".
"Being out of the EU but a member of the single market would mean complying with the EU's rules and regulations that implement those freedoms, without having a vote on what those rules and regulations are.
"It would mean accepting a role for the European Court of Justice that would see it still having direct legal authority in our country.
"It would to all intents and purposes mean not leaving the EU at all," she added.
While she also singled out the Common Travel Area between Ireland and the UK.
She said: "Maintaining that Common Travel Area with the Republic of Ireland will be an important priority for the UK in the talks ahead.
"There has been a Common Travel Area between the UK and the Republic of Ireland for many years.
"Indeed, it was formed before either of our two countries were members of the European Union.
"And the family ties and bonds of affection that unite our 2 countries mean that there will always be a special relationship between us.
"So we will work to deliver a practical solution that allows the maintenance of the Common Travel Area with the Republic, while protecting the integrity of the United Kingdom’s immigration system."
According to Independent MEP Marian Harkin, Mr Barnier told them he understands the implications of Brexit for Ireland - and that he is well-informed about the situation north and south of the border.
"One of the things he said was in his role he said there'll be no threats, there'll be no recriminations, there'll be no punishment - but he said also there will be no naivety.
"And just as Theresa May said today she wants the best deal for the UK, Barnier is saying equally (and) forcefully: we want the best deal for the EU 27".
Additional reporting: EuroParlRadio/IRN
