Theresa May has urged EU leaders to create a “dynamic” in Brexit talks that “enables us to move forward together”, at a working dinner in Brussels.
The UK PM wants to move onto trade talks but her fellow EU leaders are expected to say on Friday there has not been enough progress in negotiations.
But they may agree to begin talking among themselves about trade.
Mrs May, excluded from Friday’s meeting, told leaders on Thursday that “firm progress” was being made.
A source told the BBC the prime minister told them there was an increasing feeling “that we must work together to get to an outcome that we can stand behind and defend to our people”.
EU leaders have gathered in Brussels for a crunch summit to assess the progress made so far in Brexit negotiations with the UK, which is due to leave the EU in March 2019, following last year’s referendum result.
While Mrs May attended on Thursday, she will leave early on Friday, when the other leaders discuss Brexit without her.
They are expected to officially conclude “insufficient progress” has been made in negotiations over citizens’ rights, the UK’s financial obligation and the border in Northern Ireland to allow them to move onto the second phase of talks which will deal with trade discussions.
On Thursday, German Chancellor Angela Merkel said there were “encouraging” signs of progress in Brexit negotiations and suggested trade talks could begin in December – when EU leaders are next scheduled to meet.
She said the process was progressing “step by step” despite British media reports that negotiations were not advancing.
“I have absolutely no doubt that if we are all focused – and the speech in Florence made a contribution towards that – we can achieve a good result,” she said.
“From my side there are no indications at all that we won’t succeed.”
Over a working dinner of gnocchi and “pheasant supreme” on Thursday night, the prime minister told fellow EU leaders she was determined the UK would be a strong partner on issues from security, defence and climate change to trade, a senior government source told the BBC’s Laura Kuenssberg.
The PM told them the EU-UK’s future relationship should be a “close economic partnership” which supports “prosperity for all our peoples” and that they could be optimistic and ambitious as they already shared “the same set of fundamental beliefs”.
By BBC Europe correspondent Kevin Connolly
Theresa May issued a powerful call to the 27 EU leaders to join her in creating a Brexit deal that she will be able to stand behind and defend.
She told them she had listened both to them, and to voices in Britain, before offering money and movement on crucial issues in her Florence speech last month.
Most of the leaders said little or nothing as they left – the evening ran late after a difficult debate on Turkey – but those who did, like Angela Merkel, acknowledged the progress made so far while continuing to insist on more.
The summit won’t look like a disaster for the UK’s negotiators – they will get a public indication that the EU is preparing for trade talks whenever they might come.
But attention now shifts to the December summit, and the British need for a breakthrough there grows greater with every passing week.
She urged a “joint effort and endeavour” when her fellow EU leaders agreed on their approach on Friday and said the “urgent imperative must be that the dynamic you create enables us to move forward together”.
The other leaders are not expected to respond to her statement and Mrs May was not expected to offer anything new on the issue of the so-called divorce bill, a key sticking point.
BBC Europe editor Katya Adler said all EU leaders knew Mrs May was in a politically difficult situation and did not want her to go home empty handed, so had promised they would start talking about trade and transition deals among themselves, as early as Monday.
This could allow formal negotiations to begin around Christmas, if EU leaders deem sufficient progress has been made in discussions on other issues.
Meanwhile a group of pro-Brexit Tory and Labour politicians has urged Mrs May to walk away from negotiations this week if the EU does not accommodate the UK’s wishes.
Labour’s Brexit spokesman, Sir Keir Starmer, said it would be “irresponsible” to threaten to walk away with the talks only at “phase one”. Sir Keir and Labour leader Jeremy Corbyn are also in Brussels for their own talks.