British Prime Minister Theresa May's government fears that, unless European Union leaders offer some concessions at their summit today, the entire process of Britain's orderly separation from the EU could collapse, with grave and far-reaching consequences not only for the British economy, but also for the continent as a whole.
Yet although Germany has signalled its readiness for a potential compromise agreement today, the negotiators for Brexit, as the process of Britain's separation from the EU is now universally known, remain gloomy about the chances of a major diplomatic breakthrough.
The biggest dispute is over what should be negotiated and when, with both sides sticking to positions which are logical, but ultimately irreconcilable.
The EU argues that it cannot discuss how much access Britain should retain in European trade markets post-Brexit unless it knows first how much the British are willing to continue paying for their existing financial commitments, and unless the status of the 3.5 million EU citizens currently residing in Britain is settled. And the British, in turn, argue that they cannot say how much they can pay into the EU coffers before they know what kind of a future trading status they will enjoy with the EU.
Ultimately, this is a political tussle in which the European Commission, the EU's executive body tasked with the daily management of the Brexit talks, holds the upper hand, since it is the British who desperately need the deal wrapped up in not more than a year, so that it will be effective by March 2019, when under current treaties Britain is compelled to leave the EU, with or without a negotiated arrangement.
The British are resentful of the fact that they have published no less than 10 policy papers outlining their stance on key European trade, financial and political matters, only to have these dismissed as "too vague" by the EC.
And Mrs May, who devoted an entire speech recently to a pledge to "honour commitments" Britain has made to the EU by paying its bills well into the future, is miffed by what she sees as deliberate dithering from a Commission which simply pocketed her pledge, but still continues to allege that Britain is refusing to discuss financial arrangements.
Earlier this week, Mrs May flew to Brussels for dinner discussions with EU Commission President Jean-Claude Juncker to try to impress upon him the limited room for manoeuvre she has. Many of the anti-EU MPs within her ruling Conservative Party in London do not even accept the principle that Britain should pay anything to the EU in the future, and are increasingly arguing that leaving the EU with no deal whatsoever is preferable.
But although the discussions between Mrs May and Mr Juncker were cordial and the two "agreed that their efforts should accelerate over the months to come", as the official account of the talks claimed, chances are high that the Prime Minister will not get the concession she needs most, which is a move towards discussing the substantive shape of Britain's future relationship with the EU.
Still, British officials are hopeful that Germany may come to their rescue. For an internal policy paper which originated from the German Foreign Ministry and is now making the rounds among other German government departments claims, according to media leaks, that Germany "shares the UK's desire to secure a close partnership with the Union after its exit that covers economic and trade relations", and that the Germans want a future free trade deal with Britain to be "balanced, ambitious and far-reaching", with a high degree of access to the EU's single market.
That should be music to Mrs May's ears, although it does not address her immediate predicament, which is that German Chancellor Angela Merkel continues to insist, as do all her EU colleagues, that financial disputes with London must be resolved first.
Some observers in the British capital are now arguing that, instead of seeking to avert it, an actual breakdown in the talks may potentially be the only way out of the current impasse. For it will only be after such a breakdown that Mrs May would be able to persuade her restless Parliament to approve an EU "divorce bill" which may amount to around €22 billion (S$35.3 billion). And it would only be in response to a breakdown that the EU would be able to forge the necessary consensus to decide what relationship it is prepared to offer Britain.
For the moment, Mrs May considers this option too risky. But she may change her mind if today's summit yields no progress.