The British Prime Minister Theresa has said the Good Friday Agreement is not open to any re-negotiation.
It followed earlier reports that the deal could have been altered in order to alleviate fears over a backstop in the event of a no-deal Brexit.
Reports over the weekend suggested Mrs May was looking for a bilateral treaty with Ireland that would have meant the backstop could have been left out of the withdrawal deal.
The withdrawal agreement reached between the UK and the EU was defeated in the House of Commons by a record margin last week.
Mrs May has presented an alternative strategy on Brexit, telling the House of Commons that she will not extend the Article 50 exit process - nor will there be a second referendum there.
She said: "There are others who think that what we need is more time, so they say we should extend Article 50 to give longer for parliament to debate how we should leave and what a deal should look like.
"This is not ruling out no deal, but simply deferring the point of decision.
"And the EU are very unlikely simply to agree to extend Article 50 without a plan for how we are going approve a deal.
"So when people say 'rule out no-deal' the consequences of what they are actually saying are that if we in parliament can't approve a deal we should revoke Article 50.
"I believe this would go against the referendum result and I do not believe that is a course of action that we should take, or which this house should support.
"Second, all the opposition parties that have engaged so far – and some backbenchers - have expressed their support for a second referendum.
"I have set out many times my deep concerns about returning to the British people for a second referendum. Our duty is to implement the decision of the first one.
"I fear a second referendum would set a difficult precedent that could have significant implications for how we handle referendums in this country - not least, strengthening the hand of those campaigning to break up our United Kingdom.
"It would require an extension of Article 50. We would very likely have to return a new set of MEPs to the European Parliament in May.
"And I also believe that there has not yet been enough recognition of the way that a second referendum could damage social cohesion by undermining faith in our democracy."
'No hard border'
The backstop arrangement would kick in in the event of a no-deal scenario, to ensure there was no border between the Republic of Ireland and Northern Ireland.
Any changes to the withdrawal agreement would also need the approval of the EU, including the Irish Government.
On the next steps forward, Mrs May said: "Member of this house, predominantly but not only on the government benches and the DUP, continue to express their concern on the issue of the Northern Ireland backstop.
"All of us agree that as we leave the European Union we must fully respect the Belfast Agreement, and not allow the creation of a hard border between Northern Ireland and Ireland - nor indeed a border down the Irish Sea.
"And I want to be absolutely clear in the light of media stories this morning: this government will not re-open the Belfast Agreement, I have never even considered doing so and neither would I".
"With regard to the backstop, despite the changes we have previously agreed there remain two core issues: the fear we could be trapped in it permanently, and concerns over its potential impact on our union if Northern Ireland is treated differently from the rest of the UK.
"So I will be talking further this week to colleagues, including in the DUP, to consider how we might meet our obligations to the people in Northern Ireland and Ireland in a way that can command the greatest possible support in the house.
"And I will then take the conclusions of those discussions back to the EU".
WATCH LIVE: PM @Theresa_May makes a statement on Brexit in @HouseofCommons https://t.co/f2qY99RWTq
— UK Prime Minister (@10DowningStreet) January 21, 2019
