The prospects of that deal are looking very slim, given the chilly response from German Chancellor Angela Merkel and Irish premier Leo Varadkar.
But if no agreement can be made, Britain is either heading out of the EU on 31 October without a deal or an extension to Brexit will be granted.
Does the UK have to request an extension?
Under the terms of the so-called Benn Act, the Prime Minister - or an official in his place - has to write to the EU requesting an extension if no deal has been reached by 19 October.
Yet Boris Johnson’s aides are working on a plan to get around the Benn Act, including possibly sending a second letter “undoing” the first request.
If an extension is requested, will the EU grant one?
This is the crucial question - because there are many conflicting voices inside Europe arguing both sides. When the first extension was granted from 29 March to 31 October, it was made clear there would be no further time given.
France, in particular, is keen for the UK to get out as soon as possible so the EU can move on from Brexit. If Johnson tries the “two letters” plan, the EU could ignore the second.
Alternatively, the UK could persuade just one member state to veto the extension request and it would be thrown out.
What are the benefits for the EU of an extension?
Because it would avoid - or lessen the chances of - a no-deal Brexit. The EU would prefer to strike a deal with the UK as it would be better for trade on both sides.
Ireland’s economy, which has recently seen healthy growth, is likely to suffer from higher tariffs and other effects of a no-deal, which in turn would affect the wider EU. This week the Irish government set aside €1.2 billion (£1.08bn) to brace the country for a no-deal Brexit.
There is also an argument that, given the UK has already compromised with its double-border proposal - even though this is not yet acceptable to the EU - a final deal could be close and a few more weeks would clinch it.
There is also a chance that extending beyond 31 October could allow a second referendum, which could end up with the UK voting to remain in the EU, which would be seized on as a great victory for Brussels. This week the president of the European Parliament, David Sassoli, suggested making a general election or a second referendum a condition of extension.
What are the disadvantages for the EU in extending?
Given Brexit has been an intractable subject for more than three years, a further extension may only prolong negotiations that never lead anywhere.
French president Emmanuel Macron has led voices against an extension because he wants the EU to move on from Brexit, and earlier this week set Johnson a deadline of this weekend for an overhaul of his new proposals that would be acceptable to the EU.
The EU desperately wants Brexit to be over so it can concentrate on other priorities that have had to take a back seat - greater co-ordination, the eurozone, and EU budget talks next year.