Negotiators went behind closed doors for intensive talks in Brussels after Johnson outlined a new set of divorce terms to Irish Prime Minister Leo Varadkar on Thursday.
But they have very little time left to succeed.
EU leaders will meet in Brussels on Thursday and Friday for a summit held under the pressures of the October 31 Brexit deadline just two weeks away.
They would ideally like to have a full proposal to vote on by then.
But the sides are trying to achieve in a few days what they had failed to in the more than 3 years since Britons first voted to leave the European Union after nearly 50 years.
German Chancellor Angela Merkel will discuss the available options with French President Emmanuel Macron late Sunday and Johnson is to brief his cabinet at lunchtime.
European officials said the bloc's Brexit negotiator Michel Barnier will also brief EU ambassadors on the state of play on Sunday evening.
The few details that have leaked out suggest a compromise around the contentious Irish border issue that keeps Britain's Northern Ireland partially aligned with EU customs rules.
Whether such a fudge suits both Brussels and the more ardent Brexit backers in parliament who must still approve a deal should become clearer in the coming days.
Johnson rose to power in July on a promise not to extend Brexit for a third time – even for a few weeks.
Breaking that pledge could come back to haunt him in an early general election that most predict for the coming months.
"Getting Brexit done by 31 October is absolutely crucial," Johnson reaffirmed in a statement on Sunday.
Ireland's Varadkar hinted on Thursday that he could support the talks running on up to the October 31 deadline if a deal seemed within reach.
But Britain will only avoid a chaotic breakup with its closest trading partners if the agreement is also passed by the parliament in London – something it has failed to do 3 times.
Johnson heads a minority government and must rely on the full backing of not only his own fractured Conservatives but also Northern Ireland's small Democratic Unionist Party.
DUP's parliamentary leader Nigel Dodds warned Johnson that "Northern Ireland must remain entirely in the customs union of the United Kingdom" and not the EU.
"And Boris Johnson knows it very well," Dodds told Italy's La Repubblica newspaper.
The comments do not necessarily rule out DUP support.
UK media are presenting Johnson's mooted compromise as a "double customs" plan that could be interpreted to mean that Northern Ireland is leaving EU rules.
But the details are still under discussion and the prime minister's allies are urging lawmakers to give the British leader a chance.
The government's parliament representative, Jacob Rees-Mogg, a leading member of the Conservatives' anti-EU lobby, insisted that Johnson has the Brexit credentials to succeed.
"I think he is somebody that even the most arch-eurosceptic, even a member of the Brexit Party, can trust and have confidence in," Rees-Mogg told Sky television.
Main opposition Labour Party leader Jeremy Corbyn signalled on Sunday that he would wait for the outcome of the EU summit before trying to force an early election.
"We will look at any deal that comes up before we trigger an election," Corbyn told Sky.
Johnson is under parliamentary orders to seek a Brexit extension if no deal emerges by Saturday.
He has promised to both follow the law and yet get Britain out by October 31 – a contradiction that might end up being settled in court. – Rappler.com