British Government negotiators have been able to ascertain that the EU's actual opening position is around €60b.
"The 27 member states say they can't knock off bits of their 'bill' until the very end but politically we can't move on money until the 27 start to show compromise. As a negotiation process, it just doesn't work," said a senior Whitehall source.
"We know their position is €60b, but the actual bottom line is €50b. Ours is closer to €30b, but the landing zone is €40b, even if the public and politicians are not all there yet," the source added. A second Whitehall source confirmed the British bottom line was "€30b to €40b", while a third source said the figure in Prime Minister Theresa May's mind was "north of £30b". Making such a large offer is likely to prove politically divisive for the Prime Minister. Ardent Brexiteers will find it hard to accept paying such a sum - and certainly if it is not in exchange for a workable future relationship.
Just three weeks ago, Boris Johnson, the Foreign Secretary, told MPs in the House of Commons that European leaders could "go whistle" if they expected Britain to pay an "exorbitant" divorce bill for withdrawing from the European Union.
British negotiators are pessimistic about the chances of a breakthrough at the next round of talks, which open on August 28. The stand-off last month led Michel Barnier, the EU's chief Brexit negotiator, to warn last month that it was now "very unlikely" that talks can progress in October as planned unless there is significant movement.
A series of position papers setting out Britain's approach to Brexit on a new customs agreement and a solution to the Irish border issue could start to be published as soon as next week to give the talks some momentum. Downing Street advisers have privately warned City bosses that a British walkout from the talks is a real possibility if the impasse cannot be broken.
At the heart of the dispute is the EU's demand that the UK agree a "methodology" on the bill before moving on to trade talks - but British ministers are adamant this is tantamount to agreeing a final bill and they cannot sign cheques except as part of a final deal.
Barnier and the European Commission say the EU negotiating mandate means they cannot shift from their position, but some EU capitals have privately indicated that they want to see the talks move forward in October. Breaking the deadlock would require a formal declaration from the UK that it wanted to request a transition deal of two to three years, in which it would pay full budget contributions in return for access to the single market for a fixed period after Brexit. British officials and EU experts argue that the UK paying full contributions for up to three years could open the door to trade talks by removing the short-term budget problems caused by Brexit. Philip Hammond, the Chancellor of the Exchequer, has been lobbying Cabinet for just such a "stand-still" agreement, which the EU insists will also require the UK to accept continued free movement of people and the ongoing observance of all EU rules and regulations. EU officials insist, and UK sources concede, that a transition deal will not settle the bill issue. It would not satisfy EU demands to resolve pension liabilities for EU staff, the UK's share of liabilities to outstanding loans to Ukraine, Romania and Ireland, and past commitments to other EU projects outside the annual budget.