May and European Commission President Jean-Claude Juncker could meet early on Friday to seal a border deal, the European Commission's chief spokesman said.
The Prime Minister is under pressure to make progress in talks with Brussels amid an impasse over the border.
"There are still matters there that we would have liked to have seen clarified, we ran out of time essentially, we think that we needed to go back again and talk about those matters but the prime minister has chose to go to Brussels in relation to this text and she says she has done that in the national interest", said Ms Foster.
A European Commission spokeswoman said a news conference will be held after the leaders' meeting.
The UK prime minister said that a financial settlement which is "fair for the British taxpayer" has been agreed with the EU.
European Council President Donald Tusk revealed last night he would give a speech to media at 6:50am today but gave no indication of what he would say, beyond an update on Brexit.
All sides say they want to avoid a return to a hard border between European Union member Ireland and the British-ruled province of Northern Ireland, which might upset the peace established after decades of violence.
The DUP objected to what is known as "regulatory alignment" between Northern Ireland and the Republic, which the party claimed would mean maintaining a soft border and a new frontier with the United Kingdom mainland in the Irish Sea.
"Specifically, more work is needed around the areas of cooperation where it would be necessary to have alignment of rules and standards", the DUP said, adding that more needed to be done to establish how alignment could be effected without staying in the EU's single market and customs union.
Fifth, we will uphold the commitments and safeguards set out in the Belfast Agreement regarding North-South Co-operation.
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 United Kingdom and the European Union after Brexit. "It's an ongoing process", the spokesman told reporters.
"Ultimately if there is an agreement she will go over", he said.
That's why the two-year transition that May seeks is key - businesses want to know how long they have to plan for the future, whether that means relocation or continued investments.
"It is very, very important that whatever happens now, whatever we agree, has got to be consistent with taking back control of our laws, of our borders and of our cash", he said.