The BBC’s Laura Kuenssberg has been told there are “serious ideas” on the table that the different parties are broadly content with.
Additional wording has been added to reassure the DUP, whose opposition on Monday led to talks breaking down.
UK PM Theresa May could travel to Brussels early on Friday if a deal is reached.
European Council President Donald Tusk is due to make a statement at 0650 GMT, prompting speculation that a deal is close.
European Commission spokesman Margaritis Schinas tweeted: “We are making progress, but not yet fully there,” adding: “Tonight more than ever, stay tuned.”
But a Democratic Union Party source urged caution, saying the team were “still working”.
All sides want progress on the issue ahead of a crucial summit next week, so talks can move on to the future relationship between the UK and the EU after Brexit.
What happens to the border between Northern Ireland and the Republic of Ireland has been among the key sticking points in Brexit negotiations.
On Monday, the DUP – whose support the UK prime minister needs to win key votes in Westminster – objected to draft plans drawn up by the UK and the EU.
They included aligning regulations in Northern Ireland with those in the Republic so as to avoid border checks.
The DUP insists it will not accept any agreement in which Northern Ireland was treated differently from the rest of the UK.
The Republic of Ireland, on the other hand, which is an EU member, wants a guarantee that there will be no hard border between it and Northern Ireland after Brexit.
The UK, which is due to leave the EU in March 2019, wants to open talks on a new free trade deal as soon as possible.
The EU will only agree to discuss this when it judges that enough progress has been made on the “separation issues” – the “divorce bill”, expat citizens’ rights and the Northern Ireland border – that have been the subject of negotiations so far.
So the UK is trying to settle the Northern Ireland border issue before EU leaders meet next week. (Courtesy BBC)