The chancellor has been pushing for a transition period of longer than two years that maintains the status quo, while the foreign secretary has been battling to reduce the length of the transition and is unhappy with signing up to new European Union regulations after March 2019.
But in a sign of the circular arguments that have characterised the negotiations so far, British Brexit minister David Davis linked any settlement of the multibillion-euro bill to reaching a deal on future relations.
"We look forward to seeing how you can get it back over the net to continue this conversation", he added, emphasising that Britain wanted "to build a deep and special partnership with our friends in the EU".
"For me, we can not discuss a transition period without reaching a preliminary agreement on an orderly withdrawal. Time is of the essence", said Matti Maasikas, the Estonian EU affairs minister, whose country holds the rotating presidency of the 28-nation bloc.
Remarking that it had been six months since the United Kingdom triggered Article 50, he said progress on these three fronts was essential to allow talks to move on to the future of the bilateral trade relationship, as the United Kingdom would like.
In a bid to unblock the divorce talks, along with her words on settling budget commitments, May also offered a concession to European Union concerns that the rights of their 3 million citizens living in Britain would be protected after Brexit.
The Estonian minister who chaired a preparatory meeting on Monday for that summit said during a news conference with Barnier that the 27 were entirely united and insisted on the negotiations going through Barnier.
The third condition is the fate of Northern Ireland's border with the EU-member Republic of Ireland where both sides are wary of endangering the Good Friday peace process.
However, London hopes that progress in the talks can persuade leaders when they next meet in mid-October to give a green light to talks on future trade.
May's cabinet is now divided between a pro-EU faction - led by finance minister Philip Hammond - and foreign minister Boris Johnson who is pushing for a cleaner break, confident that the Britain can prosper on its own.
On Friday, he praised a speech by May in which she set out her plan for a roughly two-year transition period after Brexit.But us reported that Johnson had set out a new set of demands, reviving talk of a split among May's senior ministers which has the potential to destabilise her minority government. The Labour leader, Jeremy Corbyn, won young voters over in large numbers during the election.
"Some of it we'd been designing months ago".