Sinn Féin's Martin McGuinness said it was made clear to the British Prime Minister in Downing Street that Northern Ireland's Executive wants no return to a hard border on the island of Ireland when the UK leaves the European Union.
Theresa May met the leaders of the devolved administrations of Northern Ireland, Scotland and Wales in Downing Street for two hours today.
Speaking to reporters afterwards Mr McGuinness said: "Agreements like the Good Friday Agreement and the St Andrews Agreement are contingent on there being no obstructions between North and South.
"You can now travel from the centre of Belfast to the centre of Dublin in almost an hour and a half; you won't be stopped at a red light or a checkpoint of any description. We want that to continue," he added.
Northern Ireland's First Minister and DUP Leader Arlene Foster said the prime minister was very clear and definitive on the issue of a hard border.
"Theresa May had said there would not be a hard border - this an agreement between her, the Northern Ireland Executive and the Irish Government."
Mr McGuinness added that it is critical to get the European Union to agree with this understanding about Ireland's border situation.
Scotland's First Minister Nicola Sturgeon said she was frustrated that more details around the UK's negotiating position were not disclosed during today's meeting.
She also said that she would not be prepared to stand back and see Scotland "driven off a hard Brexit cliff."
Last June a majority of voters in both Scotland and Northern Ireland voted to remain in the European Union, while the Welsh electorate, by a 52.5 to 47.5 margin supported leaving.
At today's discussions the Welsh Assembly was represented by First Minister Carwyn Jones.
He is a member of the Labour party and was involved in the Remain campaign during the Referendum.
Mr Jones said this afternoon that during today's meeting he said his priority was to maintain for Wales access to the EU's Single Market.
Mr Jones added that there had been no details about what happens next in the divorce talks, and warned "time is not on our side".
Ms May offered to involve Scotland, Wales and Northern Ireland in regular formal talks on the Brexit process in an effort to quell concern about her handling of the situation.
But Downing Street said leaders of the devolved administrations must not "undermine" the negotiations.
"We have been very clear that we should be working together to secure the best possible deal for the whole country," the Prime Minister's official spokeswoman said.
"We expect representatives of the devolved administrations to act in that way and to in no way undermine the UK's position."
Following the meeting, Ms May said: "Working together, the nations of the United Kingdom will make a success of leaving the European Union - and we will further strengthen our own unique and enduring union as we do so.
"The great union between us has been the cornerstone of our prosperity in the past - and it is absolutely vital to our success in the future.
"The country is facing a negotiation of tremendous importance and it is imperative that the devolved administrations play their part in making it work," she added.
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