The EU's chief Brexit negotiator has said the British Prime Minister's speech last week had created a new dynamic in talks but warned there is still work to be done to permit the EU to move to the second phase of the negotiations.
Meanwhile, a draft resolution by the European Parliament has said that for there to be no physical infrastructure on the Irish border, then Northern Ireland would have to remain in the single market and customs union.
Following months of deadlock on the key issues in the Brexit negotiations, there were signs of a better atmosphere, if not an actual breakthrough.
At a news conference, Brexit Secretary David Davis said he was leaving Brussels more optimistic that a breakthrough could be achieved on the issues of Britain's financial liabilities, the rights of EU citizens and the issue of Northern Ireland.
However, EU chief negotiator Michel Barnier struck a more cautious note.
He said: "We have had a constructive week, yes, but we are not yet there in terms of achieving sufficient progress. Further work is needed in the coming weeks and months."
It is understood there was progress on the Irish question, particularly on the Common Travel Area.
The Government is also likely to be encouraged by David Davis's acknowledgement that these were complex issues - an acknowledgement Dublin feels has been lacking on London's part.
Last night, senior European Parliament officials adopted a draft resolution which stated that the only way to avoid a hard border in Ireland was for the UK to remain in the single market and customs union or that Northern Ireland stays in some form in the internal market and customs union.
That resolution will be voted on next week in Strasbourg.
It will only have a persuasive effect on the negotiations, but it may keep pressure on the British side to produce what are termed flexible and imaginative solutions.
The British Prime Minister last week made some concessions to EU demands in the hope of unblocking the three-month-old talks and paving the way for the EU to open discussions on a post-Brexit free trade deal.
Following Mrs May's proposal of a two-year transition in which Britain will continue to pay into EU coffers after Brexit, Mr Davis said the talks in Brussels had made "decisive steps" forward.
Mr Davis said: "We have made important progress and capitalised on the momentum created by the Prime Minister's speech.
"We are working quickly through a number of complex issues but there remain some points where further discussion and pragmatism will be required to reach an agreement."