Negotiators went behind closed doors for intensive talks in Brussels after Johnson outlined a new set of proposals to Irish Prime Minister Leo Varadkar on Thursday.
They have very little time left to succeed.
EU leaders will meet on Thursday and Friday for a summit held under the pressures of the October 31 Brexit deadline just two weeks away.
The 27 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 three 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.
European officials said the bloc's Brexit negotiator Michel Barnier will also brief EU ambassadors on the state of play on Sunday evening.
Downing Street said Johnson told his cabinet to brace for a cliff-hanger finish in the coming days.
Johnson reiterated "that a pathway to a deal could be seen but that there is still a significant amount of work to get there and we must remain prepared to leave on October 31," a Downing Street spokesman said.