Heading into the first proper round of Brexit talks on Monday, the EU and the U.K. are locked in a risky game of chicken over Britain's exit bill. But the stakes and possible resolutions to this standoff are becoming clearer as well.

The fight over what's known in official jargon as the "single financial settlement" can't last long for the talks to succeed. The EU's chief negotiator, Michel Barnier, warned last week that it could paralyze a process that's already behind schedule.

While the Brexit tab isn't as critical to Britain's post-EU economic prospects as its future trade ties with the bloc, it's now the most politically resonant issue, seen at home as a test of Prime Minister Theresa May's mettle and on the Continent as a test of the U.K.'s willingness to bend to agree the whole divorce package.

Both sides agree on one thing: not to mention any specific figure. But numbers have floated around among those inside the Brussels bubble and officials in national capitals who are working the Brexit file and struggling to come to grips with the financial arrangements of this historically unprecedented separation.

Barnier and his team have worked out and shared a methodology for computing the U.K.'s financial obligations. As in any divorce in which one side feels badly jilted, it is designed to keep London on the hook long after it officially walks out — not only to plug a looming hole in the EU budget, but also to ensure the U.K. cannot back out too soon from projects it agreed to finance as a stakeholder in the European Investment Bank.

By the EU's calculation, the single settlement, depending on a host of variables, is a figure stretching to at least 11 figures, possibly 12 — numbers large enough to set off rage and fury among some Brexiteers across the Channel, where some officials and advocacy groups believe the U.K. should be able to walk away without paying one more cent (let alone a penny or a pound).

EU officials are confident that neither May nor her top negotiator, David Davis, truly believes Britain can walk away and pay nothing.

At the low end of the EU price sheet, the consensus is €65 billion, an estimate largely based on Britain's current share of the EU's common budget (roughly 12.5 percent) through to 2020, the final year of the bloc's current long-term budget that the U.K. approved.

Others say the number will ultimately climb to €100 billion or more, taking into account continuing and yet to be determined liabilities that can only be expected to increase not shrink. One senior official said that Brussels, just before the British snap election on June 8, shared a document with the 27 capitals that suggested €80 billion — more or less — would buy the U.K.'s way through the exit turnstile.

That number is already a few weeks old. In more recent meetings in Brussels, including a session last week at the European Council, the focus has been on questions related to the European Investment Bank and other complicated financial entanglements that exist outside the EU's annual budget, which would push the U.K.'s bill higher.

Publicly, however, no one will acknowledge that any firm number exists.

On Barnier's bible

While Barnier has publicly demanded the U.K. acknowledge its obligations and put forward its own methodology, behind the scenes he has issued what is effectively a Biblical command to his team, to the 27 heads of state and government he represents and all of the people who work for them, as well as to officials in the European Council and the European Parliament: Do not give a number.

Margaritis Schinas, the chief spokesman for the European Commission, bluntly reaffirmed this edict Friday. "We shall not start the negotiations by discussing numbers," Schinas said. "Numbers will emerge once methodology is agreed. Barnier and team are ready to start ASAP."

Meanwhile, Prime Minister Theresa May's negotiating team has so far refused to describe any methodology, much to the dismay of Barnier, who complained publicly Wednesday about the unwillingness by some in London to acknowledge even the most basic obligations after more than 40 years as an EU member.

U.K. Brexit Secretary David Davis said in a statement to the House of Commons Thursday that the British government recognizes "the U.K. has obligations to the EU" and that these need to be resolved but did not elaborate on how this might be achieved.

In Brussels, the U.K. position is largely viewed as a negotiating tactic. EU officials are confident that neither May nor her top negotiator, David Davis, truly believes Britain can walk away and pay nothing.

That could be a miscalculation. There may be no issue more likely to blow up the talks than the question over the exit bill, because it goes to the heart of some of the most grievous complaints (or most pernicious myths) raised about the EU ahead of last year's referendum to abandon the bloc.

Winners and losers

Economic and financial experts, including Britain's Office for Budget Responsibility, have noted that the one-off budgetary settlement is of relatively minor importance compared to future trade deals and other commercial arrangements, as well as the U.K.'s own spending habits on services such as health care, which will be much bigger factors in the country's long-term financial performance.

There are so many variables surrounding the financial obligations of any EU member, and Britain's case is particularly complex because it not only pays into the EU budget like all other members but it also receives special reimbursements created decades ago.

Charles Grant, director of the Centre for European Reform, a London-based think tank, who is well-connected among EU officials, recently said he believes the EU's goal is a €65 billion financial settlement. In a post on Twitter, Grant said he believed a deal could be reached by the fall, by leveraging up to three transition years in which the U.K. would continue paying into the EU budget.

Other experts, however, have pushed back on the usefulness of the transition, noting that as long as the U.K. remains connected to the EU, it continues to accrue additional, potentially open-ended liabilities that only increase its ultimate bill.

In Brussels, a largely shared view among EU officials is that the U.K. will be on the hook for liabilities related to its years as an EU member for many decades to come, whether for residual costs related to satellites launched into space, or for the future pensions of EU employees.

One official estimated that the U.K.'s pension obligations alone could total €7 billion.

The other complication is how to handle the U.K.'s stake in the European Investment Bank (EIB). Discussions among the EU officials tasked with setting out Brexit negotiating positions have focused on trying to determine whether the U.K. should be forced to contribute to the so-called pooling of the EIB's assets and liabilities.

Among the options would be to have the U.K. effectively put money in an escrow account for projects that might not be completed until after the expected Brexit date in March 2019. For projects that are completed, the money would be used to pay the U.K.'s share. For projects that do not go forward, Britain could be reimbursed.

In Britain, by contrast, some analysts say the U.K.'s share in the EIB is worth more than €10 billion — and that Brussels must eventually buy back that stake, thereby reducing the Brexit tab.

EU essentials

The EU laid out its general approach to calculating the financial settlement in a position paper on "essential principles" transmitted to the U.K. last month. It said the U.K.'s burden should be based on its percentage share of payments into the EU budget — known in EU jargon as "own resources" — over the most recent, full five-year period leading up to the U.K.'s expected 2019 withdrawal.

"It's not a ransom. It's not an exit bill. It's not a punishment. It's not a revenge" — Michel Barnier

The paper stated: "The United Kingdom share of the Union obligations should be established as the ratio (in percentage) between all own resources transferred by the United Kingdom to the EU budget and the total own resources transferred by the Member States (EU28) over 2014-2018."

But the paper also laid out variables that could shift the final outcome. The EU paper specified seemingly small but crucial details, such as a demand that all obligations be calculated and paid in euros — effectively putting onto the U.K. any risk from exchange-rate fluctuations.

During a news conference Wednesday, Barnier urged the U.K. to acknowledge its obligations, rebuking British officials who have described the financial settlement as a ransom. "It's not a ransom," he said. "It's not an exit bill. It's not a punishment. It's not a revenge."

"Just a clarification; talking about financial obligations of the United Kingdom," a journalist began a question to Barnier. "Are you actually saying that as long as the United Kingdom does not in principle accept that it has financial obligations, there is no ground to talk about anything else before the negotiations?"

Barnier's succinct reply: "Yes."

Quentin Ariès, Maïa de La Baume, Charlie Cooper, Matthew Kaminski, Simon Marks and Giulia Paravicini contributed reporting.