UK Prime Minister Boris Johnson is heading for Brussels on Wednesday to meet European Commission President Ursu to discuss a post-Brexit deal.
The deadline for reaching a deal on EU trading rules is 31 December, reports BBC.
The pair will hold talks over dinner. Earlier, negotiations between officials ended in an impasse.
Major disagreements remain on fishing rights, business competition rules and how a deal will be policed.
At the dinner, Prime Minister Johnson will work through a list of the major sticking points with Mrs von der Leyen, who is representing the leaders of the 27 EU nations.
"It really is a make or break moment," said BBC Newsnight's political editor Nicholas Watt.
Any progress the two leaders make will not mean a deal is done but rather pave the way for more talks between officials, he said.
Johnson will take part in Prime Minister's Questions before travelling to the Belgian capital. EU leaders are due to meet for a summit of their own on Thursday.
A UK government source said progress at a political level may allow the negotiations - between the UK's Lord Frost and EU's Michel Barnier - to resume over the coming days.
But the source added that it was important to be "realistic" that an agreement might not be possible.
EU sources told the BBC that Barnier briefed the bloc's Europe ministers that talks were tilting towards no deal being reached before the deadline.
In separate talks on Tuesday, the UK and EU reached an agreement on specific trade arrangements for Northern Ireland - including on post-Brexit border checks and trading rules for NI, and how the new Irish Sea border will work.
It means the UK has now dropped plans to override sections of its EU exit agreement signed last year, which would have potentially broken international law.
Conservative MP Sir Roger Gale told the BBC's Newsnight he did not think Downing Street "quite understands the degree of antipathy throughout the European Union" toward the UK's plan to override the divorce deal, which he called an "expression of bad faith".
Details of the NI agreement have not yet been published, but Cabinet Office minister Michael Gove is expected to make a statement in the Commons on Wednesday.
The agreement was an important moment for businesses, said BBC Northern Ireland political reporter Jayne McCormack. "But some firms and political parties may be less receptive when they see the finer details of what's been decided," she said.
The UK and EU are at loggerheads over the so-called "level playing field" - a set of shared rules and standards to ensure businesses in one country do not have an unfair advantage over their competitors in others.
Brussels wants the UK to follow EU rules closely in areas such as workers' rights and environmental regulations, but the UK says the goal of Brexit is to break free from following common rules and reassert national sovereignty.
And the two sides disagree on how any future trading disputes should be resolved.
On fishing, the two sides continue to haggle over how much access European fishing boats should have to British waters, and how much they would be allowed to catch from next year.