Prime Minister Theresa May says she intends to return to Brussels with new ideas on the Irish backstop.
The EU has already dismissed the idea of putting a time limit on the backstop, so what other ideas have been suggested?
The backstop: a reminder
The backstop is an insurance policy - designed to avoid a hard border "under all circumstances" between Northern Ireland and the Irish Republic.
If the UK leaves the customs union and the single market that could mean goods would have to be checked as they crossed the frontier.
The UK and EU would instead like to keep the border frictionless through a comprehensive trade deal.
If such an agreement could not be reached, then to avoid those checks with customs posts or other infrastructure, the backstop would come into force.
It would keep the UK in a "single customs territory" with the EU, and leave Northern Ireland in the EU's single market for goods.
Many MPs fear the UK could be "trapped" in that arrangement for years, leaving it unable to strike its own trade deals on goods with the rest of the world.
The prime minister's Democratic Unionist Party (DUP) allies also do not want to see Northern Ireland treated differently from the rest of the UK.
So what could be done?
Clarification - and another referendum
The Irish economist Karl Whelan has suggested that the EU should clarify that it has no objections to Great Britain leaving the customs union backstop while Northern Ireland stays in it.
"This is the original version of the backstop that the EU offered, so it should be clear they are willing to still offer this," says Prof Whelan.
That would leave Great Britain free to strike trade deals but Northern Ireland would not be part of them.
That would be anathema to the DUP and other MPs.
The second part of Prof Whelan's plan is to use the Brexit political declaration to promise the citizens of Northern Ireland a referendum on the backstop, should it ever come into effect.
He suggests that five years after the beginning of the operation of a Northern Ireland-only backstop there would be a vote on whether to remain within the EU's customs union and single market.
He says: "A promise to hold a referendum five years after the end of the transition period would provide a clear concession to those who believe the backstop arrangements would be harmful to Northern Ireland by offering them a chance to convince their fellow citizens to end the arrangements after a period."
A European customs association
A paper for the German Ifo Institute suggests that the difficulties with the backstop should cause both sides to fundamentally rethink their red lines.
It proposes a new European customs association - a permanent customs union between the UK and the EU.
It would be superior to the customs deal Turkey has with the EU giving the UK "full and active participation", instead of merely being a rule-taker.
However, it acknowledges even that would not be enough to keep the Irish border frictionless and the UK would have to effectively remain in the single market for goods and perhaps services.
In return for such an enormous u-turn by the UK, the institute says that the EU should also make a radical change on free movement.
The EU's position is that the UK cannot enjoy full participation in the single market unless it accepts the four freedoms - one of which is the free movement of people.
The institute says the EU could "abandon its indivisibility dogma by which the four freedoms are inseparable, offering the UK to participate in product market integration but allowing it to make its own choices in other areas".
It adds: "Most importantly, this concerns the mobility of people."
Beef up the political declaration
The political declaration was published alongside the withdrawal deal and sets out the broad shape of the future relationship between the UK and EU.
EU leaders have said they are open to redrafting the declaration if the UK presents new ideas.
The former Liberal Democrat MEP Andrew Duff has, in a paper for the European Policy Centre, suggested improvements aimed at "rescuing" the withdrawal agreement.
He says that the language on the temporary nature of the backstop could be clearer and "more forceful".
However, given the prime minister's intention to seek legally binding changes to the withdrawal agreement changes to the political declaration are unlikely to impress pro-Brexit MPs.
Ireland leaves the EU customs union
Writing in the Daily Telegraph, the prime minister's former senior adviser Nick Timothy suggested that Ireland could leave the EU's customs union and instead create a joint customs territory with the EU.
That would eliminate the possibility of customs checks at the Irish border but mean new customs procedures between Ireland and the EU.
Mr Timothy said that would build on existing policies and would help to protect Irish consumers and businesses.
However, in Ireland there is near universal hostility to any suggestion that Brexit should force it to weaken its position in the EU.
Mr Timothy appeared to acknowledge that reality, writing: "No doubt this idea will be attacked as another unicorn."