Minister for Foreign Affairs Helen McEntee has said she believes “in the future we will have a border poll” but added that the focus for now needs to be on trying to “bring people together”.
However, he pledged to “stick 100% to the Good Friday Agreement” after he received criticism over his comments.
Speaking at an EU gathering in Co Wicklow, Ms McEntee said she wants to be “very clear” that both nations “need to be prepared” for such a vote.
She said “when the time comes we need to be ready, and ready to engage” on different questions which may arise, before saying she believes there is a need to accept that “in future we will have a border poll”.
Speaking at the same event, British Foreign Secretary Ed Miliband, who is attending the informal summit as part of a number of non-EU member state guests, declined to comment on Mr Burnham’s remarks.
However, Mr Miliband did say he wanted to make clear that the British government “are strong supporters of the Good Friday Agreement, we stand with the Good Friday Agreement”.
Mr Burnham’s remarks drew an angry response from nationalist politicians, with Sinn Féin leader Mary Lou McDonald stating that no British prime minister has the right to “high-handedly” remove the provision for a unity referendum from the Good Friday Agreement.
Asked about the comments in the House of Commons yesterday, Mr Burnham said the purpose of his visit to Northern Ireland last week was “to get functional government in Stormont”.
SNP MP for Dundee Central Chris Law put to Mr Burnham that he “has no mandate to defy the constitutional wishes of the people of Northern Ireland, the people of Scotland or the people of Wales”.
“Any written constitution or devolution agenda must include a pathway for each of those nations to choose their own future,” he added.
“I think (Mr Law) has misunderstood what I actually said,” Mr Burnham responded.
“I answered a media question on a visit to Northern Ireland before meeting the various parties and the question was that a proposal was being put to the meeting about a border poll.
“And my answer was that the purpose of the visit was to get functional government in Stormont because, in my view, that would most serve the people of Northern Ireland right now – and I wasn’t there to distract onto other issues.
“I’m very aware of every word in the Good Friday Agreement. I will stick to it 100% as Prime Minister.”
Mr Burnham added that the Good Friday Agreement was “quite clear on the circumstances” when the issue of a border poll arises.
“But our common challenge right now is functional government in Stormont and I believe everybody should work together to achieve that,” Mr Burnham continued.
He added he had asked Northern Ireland Secretary Chris Bryant “to support that process”.