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.
A senior government source told the BBC's Laura Kuenssberg the prime minister had started by saying she was determined the UK would be a strong partner on issues from security, defence and climate change to trade.
She 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 – free trade, competition, rights."
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.