Brussels rejected the U.K.'s two post-Brexit customs proposals to solve the Northern Ireland border issue.

That raises the stakes for British MPs who are revisiting the idea of staying in a traditional customs union with the EU despite the opposition of Prime Minister Theresa May and her Cabinet.

A group of senior MPs introduced a House of Commons motion Thursday calling on May's government to seek “an effective customs union” with the EU as part of the negotiation over the U.K.’s future relationship with the bloc. It will be voted on by MPs next Thursday.

The customs issue cuts to the core of the biggest obstacle to clinching an agreement on the U.K.'s orderly withdrawal: How to prevent the re-creation of a hard border between Ireland and Northern Ireland. A "backstop" proposal agreed by the two sides would effectively keep Northern Ireland inside the EU's customs union. May has declared that outcome "not acceptable" but she signed up to it on the understanding that the U.K. would be able to bring forward alternative proposals.

"So either come up with something we can work with or the backstop remains" — An EU27 diplomat

If no solution can be found, it leaves May politically vulnerable. Leaving the EU customs union in order to free the U.K. to do trade deals around the world post Brexit is a key pillar of her government's Brexit strategy. Without it, her minority administration may lose the support of Brexiteer MPs on her backbenches, potentially triggering a challenge to her leadership.

EU negotiators have made clear that they are willing to discuss terms by which Britain remains inside the bloc's customs union, or to negotiate a separate customs union like the EU's deal with Turkey, which would entail agreed-upon, common tariffs on imported goods.

But following talks on the Northern Ireland border issue in Brussels in the last two weeks, the EU has confirmed its opposition to one of the two proposals — known as the "new customs partnership." Originally put forward by U.K. negotiators last August, it would allow Britain to maintain its own system of tariffs but act as a collector of EU customs duties on goods that enter the U.K. bound for an EU country.

The other proposal, known as the "highly streamlined customs arrangement," involves using technological solutions to lower customs barriers. It is also viewed skeptically by the EU as merely a “list of best practice measures” applicable to any customs arrangement, according to one EU diplomat. That diplomat said EU negotiators do not regard it as a workable solution for the unique demands of the U.K.-EU relationship and the Irish border conundrum.

"These were already rejected before and for good reason," another EU27 diplomat briefed on the negotiations said. "So either come up with something we can work with or the backstop remains." EU chief Brexit negotiator Michel Barnier has said that if the U.K. leaves the EU customs union then some border checks are inevitable.

The continuing standoff over the future customs relationship was clear in stock-taking sessions among negotiators this week in Brussels, officials said — the first formal Brexit talks since EU27 leaders adopted guidelines on the future relationship at their March summit. A third EU diplomat characterized the discussions as having ruled out the U.K.’s "customs partnership" proposal and deemed the "highly streamlined customs arrangement" as equally unviable as a solution on its own.

In its March European Council guidelines for the next phase of negotiations, the EU warned that “Being outside the Customs Union and the Single Market will inevitably lead to frictions in trade. Divergence in external tariffs and internal rules as well as absence of common institutions and a shared legal system, necessitates checks and controls to uphold the integrity of the EU Single Market as well as of the U.K. market. This unfortunately will have negative economic consequences, in particular in the United Kingdom.”

In stating their willingness to forge an ambitious new free-trade agreement with the U.K., the EU27 leaders declared any deal should include "appropriate customs cooperation, preserving the regulatory and jurisdictional autonomy of the parties and the integrity of the EU Customs Union."

The EU's refusal to consider the customs partnership model, which would allow the U.K. to set its own duties, has taken on new urgency in London where pressure has been mounting on May to shift her red line ruling out continued membership of the EU customs union.

The House of Lords on Wednesday voted by 348 to 225 in favor of an amendment to the EU (Withdrawal) Bill calling on the government to publish a customs union strategy “outlining the steps” it had taken to secure “an arrangement which enables the UK to continue participating in a customs union with the European Union.”

Then on Thursday, a group of senior MPs introduced a motion calling for the U.K. to seek “an effective customs union” with the EU, citing the importance of avoiding a hard border on the island of Ireland. The opposition Labour Party backs the approach, as do a number of Conservative MPs — four of whom, (Nicky Morgan, Dominic Grieve, Sarah Wollaston and Bob Neill) were among the group that proposed the motion.

So far, however, May's government has not budged, insisting that it will be able to come up with an acceptable solution for the Northern Ireland border, anchored in a new customs deal.

Responding to EU diplomats' skepticism to the U.K.’s customs proposals, a U.K. official said: ‘We have put two sensible and practical solutions on the table and are working constructively towards getting this solved by October. We’re just waiting for the Commission to engage with the same spirit of cooperation."

In response to the new House of Commons motion, a spokesperson for the U.K. Treasury said: “We have been very clear that we are leaving the customs union and will establish a new and ambitious customs arrangement with the EU while forging new trade relationships with our partners around the world."

Reversing course on a customs union, a key part of the government's Brexit strategy, would be politically perilous for May. Many of the Brexiteers in her party — as well as influential members of her own Cabinet, such as Foreign Secretary Boris Johnson and International Trade Secretary Liam Fox — view Britain's ability to conduct a fully independent trade policy as a key upside of leaving the EU.

"If these positions were to evolve, the Union will be prepared to reconsider its offer" — European Council statement in March

Remaining in a customs union with the EU would restrict the U.K.'s ability to negotiate bespoke trade deals with countries around the world, and if she shifts position, May risks losing the support of the Brexiteer caucus on her backbenches.

In Brussels, EU negotiators are largely taking a wait-and-see approach, hoping that the British side ultimately gives ground and concedes that a future customs union is in Britain's own economic self-interest.

The EU has made clear it will shift its negotiating position in response to any change in the U.K.'s red lines. In its March guidelines, the European Council stated, "The approach outlined below reflects the level of rights and obligations compatible with the positions stated by the U.K. If these positions were to evolve, the Union will be prepared to reconsider its offer."