LONDON - British Prime Minister Theresa May remains optimistic about her ability to strike a deal with the European Union before the end of the week that will open the way to more substantive negotiations about Britain's trade links with Europe after the country leaves the EU.

"I am confident we can conclude this positively," Mrs May said.

The British leader may be right to remain upbeat, notwithstanding the fact that the negotiations she led on Monday (Dec 4) with EU officials in Brussels ended with no agreement.

In the words of Mr Jean-Claude Juncker, president of the European Commission, the EU's executive body, the "two sides are narrowing their positions" and hopes are high to "have a fair deal with Britain".

However, the biggest difficulty for Mrs May is not from Europe; instead, it comes from her own parliamentary backbenchers and supporters at home, who are still bickering over Britain's position towards its former EU partners.

EU negotiators have always insisted that the Brexit talks should be conducted in two consecutive stages. In the first round, matters such as how much Britain should continue to pay into the Union's coffers for existing financial obligations, border controls and what would be the status of EU citizens residing in the United Kingdom need to be settled.

Only after these are resolved could negotiations move to settling the long-term political and trade relationship between Britain and the rest of Europe.

Mrs May conceded the principle that, after leaving the EU, Britain will remain liable to pay a total of around 40 billion euros (S$64 billion) for all the special European cooperative projects it may continue to be part of, as well as the future pension liabilities of EU civil servants hired while Britain is still an EU member.

Anti-Europeans such as Mr Nigel Farage, the former leader of the UK Independence Party who still claims that Britain "owes nothing to the EU", swiftly denounced the financial agreement as a "bad deal".

But the fact that British payments to the EU will be spread over decades and are largely unquantifiable at this moment made it easier for Mrs May to push through the arrangement, and fudge the question of the final bill.

The status of the estimated 3.5 million EU citizens in the UK also appears to have been settled, with the British conceding that most will get permanent residence status.

But the biggest sticking point remains the status of the border between the province of Northern Ireland and the Republic of Ireland. It is not just the only land border between the British and the EU, but also one of deep historic sensitivities on both sides.

Most of the largely Catholic people of Ireland resent that border in the first place, since it was created by the British almost a century ago when they gave Ireland its independence. But the Protestants in Northern Ireland are determined to remain part of the UK, and insist on the maintenance of a border which contributed to many bouts of communal violence. In the last bloodshed which only ended in the late 1990s, up to 3,000 people lost their lives.

Mrs May thought she could get around the problem by promising Ireland earlier this week that the frontier "will not be a hard border", and that trade will continue to flow unhindered as it does now.

But the 10 Protestant MPs who represent Northern Ireland objected to the implication that their province will have a separate status in the UK, and vetoed the deal. And since they hold the balance of power in the parliament in London and can quickly bring Mrs May's government down, their words carry weight.

Mrs May and other British officials were reportedly scheduled to discuss the issue with the Democratic Unionist Party, the largest in Northern Ireland, on Tuesday (Dec 5).

There are ways around the problem. All of Britain could remain part of a customs union with the EU, which will diminish the need for special Northern Ireland border arrangements. And although Ireland's Prime Minister Leo Varadkar admits to be "surprised and disappointed" by the latest impasse, Britain is his country's biggest trading partner, and he has been moderate in his criticism.

Furthermore, there is a widespread feeling among other EU leaders that the opportunity to move to the substantive talks on trade by the end of this year must not be missed. Chances are, therefore, better than ever that a deal could still be hatched by the end of this week, and announced at the EU summit scheduled for next week.

Still, the experience has been a lesson for Britain. For London now knows that it will have to swallow its pride and accept whatever the EU is willing to give. And it also knows that the price for its departure from the Union would be much higher than anyone has imagined.