Theresa May is holding crunch talks with EU leaders in Brussels that could be vital in securing a Brexit deal.
The Prime Minister is meeting European Commission president Jean-Claude Juncker and European Council President Donald Tusk.
Monday marks the date that the EU gave for Britain to come up with an improved offer on the terms of Britain's withdrawal.
Without a better offer - including agreements on key preliminary issues - Mr Tusk has said he will be unable to recommend that the EU move on talks to the next stage at a summit on December 14 and 15.
Mrs May is under intense pressure to move forward the Brexit process.
But she is also facing fresh warnings from hardline Brexiteers within her own party not to make any concessions and urging her to walk away with no deal if need be.
Prominent Conservatives including Jacob Rees-Mogg, John Redwood and former chancellor Lord Lawson attempted to lay down red lines in a letter published on Sunday.
It said she should refuse to settle the "divorce bill" unless Brussels agrees to new demands including a an outline free trade deal by April 2018 and the end of free movement to the UK for EU nationals after Brexit.
Health Secretary Jeremy Hunt on Sunday urged Tory MPs to rally behind the Prime Minister, suggesting that without her there would be "no Brexit".
"The choice we face now is not between this Brexit and that Brexit, if we don't back Theresa May we will have no Brexit, and she is doing an unbelievably challenging job amazingly well," he told ITV's Peston On Sunday.
Perhaps the biggest issue she must deal with is the Irish border, with Mrs May facing demands to ensure there is no physical barrier between Northern Ireland and the Republic.
Dublin has warned they are prepared to use their veto vote to bar the start of trade talks in December if they are not satisfied, and Mr Tusk has made it clear that he will back them in any dispute over the issue.
Ireland's deputy prime minister Simon Coveney insisted that they were not seeking to delay the Brexit process.
"We certainly don't want to be vetoing anything," he told BBC One's The Andrew Marr Show.
However the influential German MEP David McAllister, who is close to Chancellor Angela Merkel, said it was proving "very, very complicated".
"Experts have been working on this issue now for months, and they still haven't come up with a solution. I would say at the moment it's still a 50-50 situation," he told Peston On Sunday.
Downing Street sought to play down the significance of the encounter, describing it as a "staging post" on the route to the full gathering with the other 27 leaders in the middle of the month.
"With plenty of discussions still to go, Monday will be an important staging post on the road to the crucial December council," a Government spokesman said.