Shadow NI Secretary Peter Kyle has said he would be prepared to call a border poll if certain conditions are met.
If Labour were in power, he said he would set out the criteria required for calling a referendum.
"I am saying I am not going to be a barrier if the circumstances emerge," he told the Sunday Politics show at the Labour Party conference.
Under the Good Friday Agreement, the power to call a border poll rests with the NI secretary.
Earlier this year, Tánaiste (Irish Deputy Prime Minister) Leo Varadkar called for clarity around the tests for when a referendum on Irish unification could happen.
There are no set criteria for when a border poll would be triggered.
The legislation states the minister shall call a referendum on Irish unity "if it appears likely to him that a majority of those voting would express a wish that Northern Ireland should cease to be part of the UK and form part of a united Ireland".
Mr Kyle said the conditions currently did not exist but he declined to elaborate on the criteria.
While constitutional issues were important, they were a distraction from the cost of living crisis, he added.
Ulster Unionist Party leader Doug Beattie said Mr Kyle's comments were a distraction from the "immediate challenges" facing Northern Ireland.
He added that they were "unhelpful and ill-timed" and he had contacted Mr Kyle to express his concern.
Sinn Féin's Chris Hazzard said the comments were "recognition from Peter Kyle that the current position is unsustainable".
"It's having your head in the sand simply to deny the fact that constitutional change is coming, that people are talking about this, civic society and political parties and all around us," the South Down MP said.
"So, it's a responsible thing to do.
"We obviously have to wait to wait to see if he even gets into post to be able to do this."
SDLP leader and Foyle MP Colm Eastwood said conversations needed to start happening.
"The most important thing for those of us who want to see change, we need to make the case," he said.
"We need to convince people, we need to talk to people over the next number of years and make sure they can feel comfortable with the change that we are proposing."
The Democratic Unionist Party (DUP) warned the Labour Party risked "playing dangerous games if they appease Sinn Féin's divisive border poll plans".
"Peter Kyle should focus on removing the protocol which has damaged east-west relations and utterly undermined devolution," said East Belfast MP Gavin Robinson.
"Previous Labour administrations were able to respect the cross-community consensus needed for progress in Northern Ireland."
Speaking at a conference fringe event on Sunday evening, Mr Kyle said he was aware his remarks were getting traction in Northern Ireland but he had "simply stated the law".
The DUP withdrew from the executive in protest over the Northern Ireland Protocol, which is part of the Brexit deal, after the Northern Ireland Assembly election in May.
Earlier this week Northern Ireland Secretary Chris Heaton-Harris warned he would call a fresh Stormont election, as obliged to under current rules, if the institutions were not re-formed by 28 October.
"It would be a sign of abject failure not just in Northern Ireland, but also of the government and the secretary of state for not being able to facilitate the forward movement," said Mr Kyle.
"But once we pass legislation and we set the rules for an election, then we really need to stick to those rules in all but extreme circumstances."
Recently there has been criticism of how much time Northern Ireland secretaries spend in Northern Ireland.
Mr Kyle, who was appointed to the shadow portfolio in November 2021, defended his own record.
"It depends how much I'm needed in Westminster," he said.
"My instinct has always been to get over to Northern Ireland as often as I can."
He said he had made two visits to Northern Ireland in the past two months.
"I take my job incredibly seriously," he said.
"These are sometimes challenges that the Northern Ireland secretary and the shadow secretary can only deal with in Westminster because I'm not a Northern Ireland politician. I represent Northern Ireland in Westminster."
Sunday Politics is available to watch on BBC iPlayer
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