Top EU officials have expressed optimism that a Brexit deal can be struck by the end of the year.
Jean-Claude Juncker, the head of the European Commission, said the chance of the UK and the EU reaching a deal has increased in the last few days and could be agreed by November.
Meanwhile, European Council President Donald Tusk said an agreement was possible by the end of 2018.
But Irish PM Leo Varadkar said there is still "a fair bit of work to be done".
The UK is due to leave the EU on 29 March 2019.
However, there is still no agreement on some issues, including how to avoid new checks on the Irish border.
Both sides had hoped to finalise the so-called divorce agreement and agree a statement on their future economic co-operation by an EU summit in 11 days' time.
Speaking to the Austrian press on Friday and asked whether an agreement could be reached at the next meeting of European leaders on 17 October, Mr Juncker said: "We are not that far yet. But our will is unbroken to reach agreement with the British government."
He said a deal could be agreed by November.
"I have reason to think that the rapprochement potential between both sides has increased in recent days," Mr Juncker added.
He also reiterated his position that a no-deal scenario "would not be good" for either the UK or the EU.
'End of the year'
Speaking on Saturday, European Council President Mr Tusk said: "We will try for it [agreeing a deal] in October... and I think there is a chance to have an accord by the end of the year."
Foreign Office minister Sir Alan Duncan also expressed optimism that a deal could be reached before December.
There were also reports from EU officials of a better atmosphere in talks over the Irish border.
The upbeat assessment of progress in negotiations prompted sterling to rise against the Euro and the US dollar.
But Mr Varadkar warned that the two sides had not crossed the finish line yet and the October summit was "a time to take stock".
He told reporters: "I would be hopeful at that point that there would be decisive progress allowing us to conclude an agreement by November.
"That remains to be seen yet. I think there is a fair bit of work to be done."
He added: "It's increasingly important that we conclude a deal sooner rather than later."
The BBC's Brussels correspondent Adam Fleming said officials still "seem to be pushing for the whole withdrawal agreement to basically be done" by the EU summit.
"Although, if we have learned anything from Brexit it's that the timetable is incredibly flexible, to use diplomatic language," he told BBC Radio 4's Today programme.
Downing Street distanced itself from suggestions - reported in the Daily Mail newspaper - that they were concerned Mr Juncker was trying to "bounce" the UK into agreeing a Brexit deal by mid-October.
Courting Labour votes
Meanwhile, the Guardian reported senior Conservatives had been in private contact with a number of Labour MPs to persuade them to back Theresa May's Brexit deal.
Some Labour MPs who were mentioned in the article took to Twitter to refute the claims. Rachel Reeves tweeted: "All the Labour MPs listed work hard and fight Tories locally and nationally every day."
It comes weeks after the head of the European Council Donald Tusk said Theresa May's Brexit plans were unworkable.
Both sides had hoped to finalise the so-called divorce agreement and agree a statement on their future economic co-operation by the October summit.
But last month, the EU's chief negotiator Michel Barnier said that if both sides were "realistic" there could be an agreement by November, when a special one-off summit has been arranged.