Relaying comments made by the EU's Brexit negotiator, Michel Barnier, at a meeting of the bloc's ministers in Luxembourg, they said that if the Tuesday deadline was missed then talks would have to continue after the Thursday-Friday summit.
British Prime Minister Boris Johnson insists his country will leave at the end of the month with or without an agreement, although United Kingdom lawmakers are determined to push for another delay rather than risk a chaotic no-deal Brexit.
London's proposed plan would see Northern Ireland remaining in the UK's customs territory but operating the EU's rules and procedures on tariffs.
Pro-Brexit Conservative lawmaker David Davis said Wednesday that success rests on the stance of Northern Ireland's Democratic Unionist Party, the ally of Johnson's Conservative government.
The negotiators are meeting again on Wednesday, but without the DUP's support, Johnson will face a struggle to get any agreement approved in a vote in the UK Parliament.
European officials said the bloc's Brexit negotiator Michel Barnier will also brief EU ambassadors on the state of play this evening.
But the sides are trying to achieve in a few days what they had failed to in the more than three years since Britons first voted to leave the European Union after almost 50 years.
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An EU official said "one of the major outstanding issues" was agreeing on Britain's application of common bloc rules and standards created to ensure fair competition.
If he is able to get an agreement, Government sources have said they will seek to rush through legislation to ratify it in time for the promised Halloween withdrawal date.
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".
"But if it's not, there is still more time".
After emerging from Number 10, Mrs Foster said: "We respect the fact negotiations are ongoing therefore we can not give a detailed commentary".
Addressing a joint press conference with the visiting Belgian Prime Minister Charles Michel, Rinne said that "all options are still possible".
But if an agreement is not ready for the Brussels summit, British Prime Minister Boris Johnson will nearly certainly have to postpone Britain's exit again.
Conservative lawmaker Owen Paterson said on October 15 that the prime minister's emerging divorce deal was "unacceptable".
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After weeks of gloom the last few days have given a glimmer of hope that an agreement can be reached, but there has so far been no decisive breakthrough.
British Brexit Minister Stephen Barclay joined Barnier in Luxembourg in what was seen as a positive sign for the talks, and said "a deal is still very possible".
He said "several complex technical issues still have to be resolved".
Mr Barnier was reported to have raised concern about the complexity of a British plan to keep Northern Ireland in the United Kingdom customs territory while avoiding the need for border controls.
"All scenarios are open", she said. Johnson is said to have given ground on customs rules for Northern Ireland in a bid to reach a formal text. But EU diplomats were pessimistic about the chances of Johnson's hybrid customs proposal for the Irish border riddle.
If he fails to strike a deal with the European Union, a law passed by parliament obliges him to seek a delay - the scenario European Union diplomats think is most likely.
Irish Prime Minister Leo Varadkar said talks were "moving in the right direction" but gaps between the sides remained.
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