The president of the European Council, Donald Tusk, has urged European Union leaders not to humiliate Theresa May when they discuss her latest request to delay Brexit.
But the other 27 EU countries (known as the EU27) may still impose tough conditions on any extension to the Article 50 negotiating period, to ensure that they protect their own interests.
So what do they have in mind?
The EU’s main initial demand is that the UK fulfils its commitment to hold European Parliament elections on 23 May, if Brexit hasn’t happened by then.
Draft conclusions for today’s summit in Brussels say that if the UK were to fail to hold the elections, it would automatically leave the EU – without a deal if necessary – on 1 June.
But even if those elections take place, and UK MEPs attend the first session of the new parliament in July, there is still concern in some quarters that a disgruntled British government could prevent the EU making progress on a series of important issues.
Different member states have different opinions on how serious a threat this is. But it is certainly an important part of the discussion.
Theresa May has already pledged that the UK will remain a “constructive and responsible” member state, and abide by its “duty of sincere co-operation” with the rest of the EU.
This is a form of words that carries weight in Brussels, but it doesn’t bind any future UK government to act in the same way.
What else is on the EU’s mind?
The EU has a series of big decisions to make in the coming months. It doesn’t want Brexit to get in the way.
To begin with, there is a changing of the guard at the top of the EU’s political structures.
A new president of the European Commission has to be nominated by EU leaders, and approved by the European Parliament, in time to take over on 1 November from Jean-Claude Juncker, who is standing down.
A month later, on 1 December, a new European Council president will take office, succeeding Donald Tusk.
There are other important issues looming for the EU, notably on the size and shape of the next seven-year budget for 2021-27.
Negotiations on the budget will start to intensify under the new Commission, although final decisions won’t be made for at least a year.
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That is one of the reasons why any Brexit extension will almost certainly, as Mr Tusk has suggested, be “no longer than a year”.
Brexit has been taking up far too much time already. It has diverted the EU from a focus on other subjects, and the leaders of the EU27 would rather concentrate on their own future than on the internal debates of a country that is trying to leave.
So the leaders will have to decide how many conditions they impose on a Brexit extension, and they will want to hear Mrs May repeat her written pledge of co-operation in person.
Throughout the Brexit process, the unity of the EU27 has been a consistent factor. But there is now a difficult balance to strike.
Does there need to be a formal review process during an extension, or is a European Council decision to “remain seized of the matter” (EU jargon for paying close attention to it) enough?
“I think the extension should be as short as possible,” German Chancellor Angela Merkel told the German parliament.
“But it should be long enough to create calm, so we don’t have to meet every two weeks to talk about the same subject,” she added.
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