EU Council president Donald Tusk said he was "encouraged by progress" and a deal on Ireland, the "divorce bill" and citizens' rights was "getting closer".
Theresa May is meeting EU figures in an attempt to finalise the deal ahead of a summit in 10 days' time.
Belgian MEP Philippe Lamberts told the BBC's Laura Kuenssberg that the UK had made a concession on the Irish border.
Mr Lamberts had said the UK was prepared to accept that Northern Ireland may remain in the EU's customs union and single market in all but name. But, she stressed, the BBC has not seen the draft document nor has it been signed off.
Downing Street sources also told her they were cautious as to whether a deal would be done in Brussels on Monday, with one source saying: "There are still moving parts".
Mrs May is meeting European Commission president Jean-Claude Juncker and Donald Tusk, the president of the European Council.
Scottish First Minister Nicola Sturgeon reacted to reports that Northern Ireland could retain "regulatory alignment" with the EU by saying there was "surely no good practical reason" why other parts of the UK could not do the same.
And Mayor of London Sadiq Khan tweeted that there could be "huge ramifications" for the capital if the country was given this deal, suggesting he would seek something similar.
While the Republic of Ireland's prime minister, Leo Varadkar, is expected to make a statement later, DUP leader Arlene Foster has said her party "will not accept any form of regulatory divergence" that separates Northern Ireland from the rest of the UK.
She went on to accuse the Republic of Ireland of "seeking to unilaterally change" the Good Friday Agreement - the peace deal that brought an end to the Troubles - without the DUP's "input or our consent".
"Of course we will not stand for that," she added.
Mr Tusk represents the leaders of the other 27 EU members, who all need to agree for there to be a move to the next phase of talks.
The UK voted for Brexit last year and is due to leave in March 2019, but negotiations have been deadlocked over three so-called separation issues.
The EU says it will only recommend the start of talks about future trade arrangements when it deems "sufficient progress" has been made on three issues - the status of expat citizens, the "divorce" bill and the Northern Ireland border.
The UK has been set a deadline of this week to come forward with an improved offer on them, and hopes that the go ahead for future talks will then be given at an EU leaders' summit on 14-15 December.
On the "divorce bill", the UK is understood to have recently increased its offer, which could be worth up to 50bn euros (£44bn).
On the issue of rights for the three million EU citizens in Britain, the UK has agreed that those who already have permanent residence will not have to pay to apply for settled status. Those making a first time application for the right to stay after Brexit, however, will face a charge - reportedly similar to the cost of applying for a passport.
Settled status will grant those who have spent five years in the UK equal rights on healthcare, education, benefits and pensions to British citizens.
Ministers have already suggested people legally resident in the UK before an as yet unspecified cut-off date will be allowed to stay and they want to make the process "as easy as renewing a driving licence".
Progress in other areas has led to attention being focused on the Ireland question in recent days.
The Irish government is seeking guarantees from the UK that there will be no customs checks on the border with Northern Ireland after Brexit and movements of goods and people will remain seamless.
Mr Tusk has stated Dublin must be satisfied there will be no return to a "hard border" and as Irish ministers met on Monday, it seemed there was yet to be a resolution.
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