Members of the European Parliament were briefed on a draft text of an agreement reached between the two sides after seven months of tough negotiations.
The EU has stuck firmly to three demands: that Britain pay a substantial "divorce bill," that rights of European citizens in the UK are guaranteed, and that there is no reinstatement of a border infrastructure between Northern Ireland, which will leave the EU with the rest of Britain in March 2019, and the Republic of Ireland.
Substantial progress on the first two issues was made last week but wrangling on the Irish border continued through Sunday night and into Monday, hours before Theresa May arrived for a working lunch with EU Commission President Jean-Claude Juncker.
The dismantling of a so-called "hard border" was a key plank of the Good Friday Agreement, which brought peace to Northern Ireland after years of sectarian conflict. The Irish government was determined to secure a cast-iron guarantee from the UK that there would be no return to border controls after Brexit.
Britain has been desperate to reach a deal on the outstanding issues to secure an agreement from the EU that it would move on to discussions about a future trading relationship.
Philippe Lamberts, leader of the Greens in the European Parliament, told CNN that he had seen draft text of an agreement in which the UK had agreed that Northern Ireland would continue to be aligned with EU laws and regulations that would otherwise require checks at the border.
When asked what was outlined with regard to Northern Ireland, Lamberts said that "it is what is needed."
"We needed a commitment to have no divergence. (The UK) didn't want that wording but what we have, it is full alignment."
Lamberts went on to say he was surprised the UK had agreed, adding that while some may see Monday's developments as "concessions" he called the move coming "to terms with what is needed."
Any draft agreement has yet to be formally announced and ratified and Downing Street has refused to comment.
While there are positive signs of progress between the two sides, May will still need the support of the Democratic Unionist Party, the hardline Northern Ireland party whose 10 Westminster lawmakers are propping up her minority government.
May will now also meet the President of the European Council, Donald Tusk, later in the day while in Brussels.
Tusk scrapped a scheduled trip to Israel, citing a "critical moment" in Brexit negotiations. An EU official said the trip was abandoned so that Tusk can be available for "consultations on draft guidelines" for potential trade discussions.
Tusk said in a tweet that he was "encouraged" on the issue of Ireland and "closer to sufficient progress" following a phone call with the Irish Taoiseach, Leo Varadkar, on Monday.
"Tell me why I like Mondays! Encouraged after my phone call with Taoiseach @campaignforleo on progress on #Brexit issue of Ireland. Getting closer to sufficient progress at December #EUCO."
Varadkar is due to make a statement on phase one Brexit discussions shortly.
The UK is hoping the EU heads of government will deem Monday's developments enough progress to give the go ahead for negotiations to move on to a second phase of negotiations at an EU summit next week. This phase would focus on a future trade relationship between the UK and EU and transitional arrangements.
A Downing Street spokesman told CNN on Monday that "good progress" was being made in Brexit negotiations, but cautioned that more needed to be determined. The UK government official added that focus is now on the EU summit on December 14.
Ireland had wanted firm assurances of Britain's intentions for the border when it leaves the EU.
"These discussions are in a sensitive place right now," Irish Deputy Prime Minister Simon Coveney told RTE Radio 1 earlier on Monday. "Both governments understand what each other is asking for, and obviously there's consultation going on with both governments to try and get that right"
Coveney, who is also Foreign Minister, seemed optimistic that the two sides would be able to strike a accord earlier on Monday but warned that "our partners across the EU will hold firm with us, if that's necessary."
"We're saying need more reassurance before we move on to phase two," he added.
The EU had given May until December 4 to put forward additional proposals that would resolve three areas of contention that have stalled discussions: Britain's financial settlement or "divorce bill"; the rights of European Union citizens in the UK after Brexit and the thorny issue of the Northern Ireland border.
In London, the EU Withdrawal Bill was once again under scrutiny in the House of Commons. On the agenda Monday, lawmakers will be discussing how leaving the bloc is going to affect other parts of the UK, not including England. There is concern from ministers that powers which once resided with the EU will go back to Westminster, rather than transfer to devolved areas.
Over the weekend, top Brexit supporters demanded that May stand her ground on "any further financial commitment to the EU until they have agreed that in return, they will meet a number of conditions."
Preempting the mid-December summit, the Leave Means Leave group sent a letter outlining several terms, including calling for Britain to be beyond the jurisdiction of the European Court of Justice (ECJ) and for no new EU regulations to apply once Britain exits in March 2019.
May has promised to end the ECJ's authority in the UK, however she has suggested its remit might continue in some capacity during an "implementation period" past March 2019.