Both sides said they were close to finding a basis for a treaty to ensure Britain heads for a managed withdrawal from the European bloc it has been part of for almost half a century.
Many European officials are pessimistic about the chances of a rapid deal, but none publicly rule it out.
If a deal is reached, British Prime Minister Boris Johnson would be able to put it to the British Parliament on Saturday and so avoid having to seek another delay beyond Oct 31.
"Even if an agreement will be hard - more and more hard to be frank - it will still be possible this week", EU's chief Brexit negotiator Michel Barnier told reporters in Luxembourg yesterday morning before a meeting with ministers from European Union governments.
Sources said the PM was considering a call to European Commission president Jean-Claude Juncker today to help speed up the process. But Northern Ireland's Democratic Unionist Party, the party that props up Johnson's minority government, strongly opposes any measures that could loosen the bonds between Northern Ireland and the rest of the United Kingdom.
A second European Union official said an agreement was "close but not 100pc certain", adding "there are still parts that need to be nailed down".
"But whether we will be able to conclude a revised withdrawal agreement, which after all is an global treaty, in time for the summit on Thursday, that's as of now unclear", he said, adding that some hours earlier the gap had been "quite wide, particularly on the issue of customs".
Differences between the two sides remained but were narrowing to some technical and complicated customs and value-added tax issues, officials said.
Brexit negotiations will resume in Brussels on Wednesday morning after "constructive" negotiations went into the night on Tuesday, a British spokesman said.
Johnson has long seen the upcoming summit as the crucial moment for Brexit talks.
Disagreements centre on a future trade deal and the rejection by Northern Ireland's Democratic Unionist Party of customs solutions tentatively agreed by negotiators for the Irish border.
Although differences over the complex divorce between the world's fifth-largest economy and its biggest trading bloc have narrowed significantly, European Union sources reported on Wednesday that the two sides had reached a "standstill".
"There is a pathway to a possible deal but there are many issues that still need to be fully resolved", Varadkar said in a speech.
Johnson hopes to get approval for the agreement in a vote at an extraordinary session of the British parliament on Saturday, to pave the way for an orderly departure on October 31.
But Mr Barclay insisted the government was still committed to leaving on 31 October and when asked how he was planning to do so, responded: 'We will cross that bridge when we come to it'. "We think the best way of doing that is with a deal, to leave in a smooth and orderly way". "There would be no customs duties on goods crossing to Northern Ireland if they were to stay there".