Theresa May has suggested she could attempt to negotiate a time limit or exit mechanism to the Irish backstop when she returns to Brussels this week.
The statement comes after MPs voted in favour of Conservative backbencher Sir Graham Brady's amendment to seek "alternative arrangements" to the backstop, which is designed to prevent a hard border on the island of Ireland.
Writing in the Sunday Telegraph, Mrs May said: "While replacing the backstop with alternative arrangements was one option, [Sir Graham] would also be happy with the current backstop if there was a time limit or unilateral exit mechanism."
Irish Tanaiste Simon Coveney, however, dismissed such suggestions, stating the backstop is required to "ensure the protection of the Good Friday Agreement".
Writing in the Sunday Times, Mr Coveney added: "The EU will not renegotiate the withdrawal agreement and there will be no withdrawal agreement without the backstop."
"However, the EU is prepared to look again at the political declaration to make it more ambitious in terms of the future relationship, if the UK so wishes.
"There is a deal on the table, and walking away from it will have serious consequences for us all."
The backstop would see the UK remain in a customs union with the EU, and Northern Ireland remain in large parts of the single market for goods, in order to prevent the need for checks or infrastructure on the border.
It is designed only to be used if no other agreeable arrangement can be found between the UK and EU before the Brexit date of March 29.
On Sunday, Mrs May also stated she is "determined" to deliver Brexit on time and the exit date will not change.
DUP leader Arlene Foster said that a deal can be agreed if the "toxicity" of the backstop is dealt with.
"I think we really need to focus on trying to get a deal. That's what the DUP want, that's what the government wants and I believe it's what the European Union wants," Mrs Foster told BBC Radio Ulster.
"I'm certainly not giving up. I think it's the right thing to do to go back and say that whilst there are other issues with the withdrawal agreement, if this issue around the toxicity of backstop was dealt with then we would be able to move forward."
While last week the Prime Minister secured the backing of the majority of MPs to return to Brussels to renegotiate, there has already been signs of dissension within her own party.
Senior Conservative Brexiteer Steve Baker warned that there is "trouble ahead" for the Prime Minister, suggesting what she will seek from Brussels to get a Brexit deal will not win her backbench support.
"Trouble ahead. Leave-backing MPs voted to support alternative arrangements in NI but with grave misgivings about the whole agreement," he tweeted on Sunday.
"Now the PM co-opts us into accepting everything but the backstop and, on the backstop, accepting a codicil."