Brexit minister David Davis underlines 'open border' aim
Brexit Secretary David Davis has said that the UK and Republic of Ireland "both want to have an open border".
Mr Davis met with First Minister Arlene Foster and Finance Minister Máirtín Ó Muilleoir separately on Thursday.
"There are other places in Europe that don't have hard borders with places outside the union," he said after speaking with Mrs Foster.
However, Mr Ó Muilleoir said he was determined that Northern Ireland would remain a part of Europe.
The Sinn Féin finance minister - who was standing in for Deputy First Minister Martin McGuinness who is on leave - described his meeting with Mr Davis and Northern Ireland Secretary James Brokenshire as a "frank exchange".
"It's my resolve and conviction that we will ensure that the Irish government and the British government get together to make sure that we are not dragged out of Europe," he said.
"That we remain at the heart of Europe and it is up to him (Mr Davis) to square that particular circle.
"But, the majority of people here voted to stay and that vote to remain should be respected."
Mark Devenport, BBC News NI Political Editor
In an article in today's newspapers, David Davis says he wants to see continuity when it comes to public funding, but Finance Minister Máirtín Ó Muilleoir will probably be pressing him to go further.
David Davis is also quoted as saying the government does not want a hard border and wants a practical solution that will work in everyone's interest here.
David Davis and Secretary of State James Brokenshire both emphasise there was a common travel area between the Republic Ireland and the UK before either was a member of the European Union.
Speaking after his meeting with Mrs Foster, Mr Davis underlined the importance of controlling immigration and said "we have to take control of our borders".
He identified the single energy market, export markets and the skills base as "things which are important to making Brexit a success in Northern Ireland".
Mr Brokenshire said it was "important" to move on from the EU referendum and focus on getting "the best possible deal for the UK and the best possible deal for Northern Ireland".
Both Mr Davis and Mrs Foster campaigned for a leave vote in the EU referendum and believe Brexit offers excellent potential trade opportunities.
DUP sources described their meeting in Belfast on Thursday as "useful".
Mr Davis has been attending the first session of a business group set up to advise the Northern Ireland Office on local concerns about the forthcoming Brexit negotiations.
In Northern Ireland, the majority of voters (56%) opted for the UK to stay in the EU during June's referendum.
At a special meeting of the cabinet on Wednesday, Prime Minister Theresa May said there would be no second referendum on EU membership.
She said there would also be no question of the UK trying to remain inside the EU through some kind of back door.
Mr Davis has said he wants to reach out to the parts of the UK where people did not support Brexit, as well as those where big majorities voted to leave the EU.
In an article in Thursday's Belfast Telegraph, he wrote that Northern Ireland's voice will be heard loud and clear and Brexit should not mean the introduction of a hard border.