British Prime Minister Theresa May said her government was looking at a number of options to resolve the concerns of MPs over the backstop proposals in her Brexit plan.
She was addressing MPs after holding cross-party talks in the wake of the overwhelming rejection of her Withdrawal Agreement by the House of Commons last week.
She said that the EU was very unlikely to extend Article 50 without a plan for a "no-deal" Brexit and that revoking Article 50 would go against the Brexit referendum result.
Mrs May also dismissed calls for a so-called People's Vote or second referendum on Brexit, saying that it would set a difficult precedent.
"I fear a second referendum would set a difficult precedent that could have significant implications for how we handle referendums in this country," she said.
The British Prime Minister also said that the Good Friday Agreement would be fully respected through Brexit and it would not be re-opened.
It followed newspaper reports this morning that the British government was considering re-examining the 1998 agreement to break the backstop deadlock.
In her statement, she also acknowledged the continuing concern about the backstop, which is designed to avoid a hard border on the island of Ireland post-Brexit.
Mrs May told MPs: "I will be talking further this week to colleagues, including the DUP, to consider how we might meet our obligations to the people of 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."
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Mrs May also announced that she had scrapped the £65 fee for EU nationals wishing to remaining in the UK with "settled status".
An Irish Government spokesperson said that it noted Mrs May's statement to parliament.
They said: "The consistent Irish and EU position is that we want to secure agreement.
"The Withdrawal Agreement and the backstop provisions are settled following extensive negotiations, but as Michel Barnier said today, if the UK position on red lines evolves, there may be scope on the EU side for a more ambitious future relationship."
Earlier, European Affairs Minister Helen McEntee said that Ireland would not engage in bilateral talks on Brexit and would only negotiate as part of the 27 remaining members of the European Union.
The Sunday Times yesterday reported that Mrs May was seeking a treaty with Ireland to remove the backstop arrangement.
Speaking on RTÉ’s Morning Ireland, Ms McEntee said: "What we can't do and what we won't do, because we have not throughout this entire process, is engage in any kind of bilateral negotiations with the DUP or any other political party in Northern Ireland or the UK.
"This is a negotiation between the EU and the UK."
Speaking on the same programme, the DUP's Jeffrey Donaldson said it is time for meaningful engagement on Brexit given the seriousness of the situation and the short time left.
He said that "we need to find solutions and a way of addressing concerns over the backstop".
Mr Donaldson said: "I don't think that relying on the possibility of the EU and UK resolving this issue, without input from Belfast and Dublin, is unrealistic."
He said the challenge is to come up with a solution that avoids a hard border and at same time avoids a border in the Irish Sea.