Boris Johnson will fly to Brussels later for talks on a post-Brexit deal with the European Commission President Ursula von der Leyen.
Time is running out to reach a deal before 31 December, when the UK stops following EU trading rules.
The pair will hold talks over dinner, after negotiations between officials ended in deadlock.
Major disagreements remain on fishing rights, business competition rules and how a deal will be policed.
At the dinner, expected to begin at 19:00 GMT, the prime minister 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.
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.
Speaking at Prime Minister's Questions, Mr Johnson said "a good deal is still there to be done", but stressed that the EU's position was to have an "automatic right to punish us" if the UK's rules and regulations didn't move in step with the EU in the future.
Meanwhile German Chancellor Angela Merkel has said a Brexit deal was still possible but insisted that the integrity of the EU single market must be respected.
EU sources told the BBC that Mr Barnier briefed the bloc's Europe ministers that talks were tilting towards no deal being reached before the deadline.
BBC political editor Laura Kuenssberg says that the purpose of the dinner is not to call a halt or to proclaim that a deal's been done.
"The reason for the meeting is to see if both sides are willing in principle to tolerate the notion of budging, after the negotiations, and frankly negotiators, have been exhausted," she says.
EU leaders are due to meet for a summit of their own on Thursday.
It comes as no surprise at all that EU-UK talks have gone down to the wire.
That was widely predicted. As was the "both making last-minute political compromises" part, believed to come right at the end, after the two sides abandon their high-stakes game of chicken.
But tonight's dinner is far more complex than the prime minister declaring "OK, Ursula, we'll give you your level playing field, if you (EU) give us our fish."
The clash of ideologies, clear from the start, is still very much present.
We have the government's determination to protect its post-Brexit sovereignty, and not sign up to another Brussels rule book on the one side and on the other, the EU focus on protecting its single market a) from what it views as potentially unfair competition from the UK and b) in terms of global reputation.
Brussels believes compromising single market rules for a UK deal, would weaken it in the eyes of other future trade partners.
Conclusion: As much as Boris Johnson and Ursula von der Leyen, say they still hope for a deal, 'no deal' is still very much on the table tonight, alongside dinner.
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.
Details of the NI agreement have not yet been published, but Cabinet Office minister Michael Gove is making 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.
Brexit - The basics
- Brexit happened but rules didn't change at once: The UK left the European Union on 31 January 2020, but leaders needed time to negotiate a deal for life afterwards - they got 11 months.
- Talks are happening: The UK and the EU have until 31 December 2020 to agree a trade deal as well as other things, such as fishing rights.
- If there is no deal: Border checks and taxes will be introduced for goods travelling between the UK and the EU. But deal or no deal, we will still see changes.
Speaking to the BBC, Mr Gove confirmed that EU officials would be present in Northern Ireland "to make sure the processes, that we are in control of, will be running appropriately".
Former Conservative leader Iain Duncan Smith said there would be a need "for some explanation" from the government because MPs had been "assured by the government that there were would be no customs officials sitting around".
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.