The UK prime minister said that a financial settlement which is "fair for the British taxpayer" has been agreed with the EU.
"It hasn't been easy for either side", May said in an early-morning news conference in Brussels following all-night talks.
EU Commission spokeswoman Mina Andreeva said on Twitter that they were "likely to meet this morning at 7:00 AM (0600 GMT) in the Berlaymont", the organisation's headquarters.
May's European Union partners insist that the talks must make "sufficient progress" on Britain's financial settlement, a way to keep open Northern Ireland's border with Ireland and the rights of citizens hit by Brexit.
"Companies all across the United Kingdom want absolute clarity on the long-term deal being sought, and want government to work closely with business experts to ensure that the details are right".
On the issue of the UK's so-called "divorce bill", which is expected to total up to £50 billion, Mrs May said that in her landmark speech in Florence in September she had made clear the United Kingdom was "a country that honours our obligations".
But the EU will only move to trade talks if there is enough progress on three key issues: the money Britain must pay to the EU; rights for EU citizens in Britain and British citizens in the EU; and how to avoid a hard border with Ireland. "We will stay very firm".
"A spokesman for PM May said Brexit discussions were ongoing while a senior Irish official said talks were moving swiftly and that a deal was possible in hours".
Barnier informed European Union ambassadors that Downing Street had told him a potential solution was being worked out that could possibly satisfy both Northern Ireland's Democratic Unionist Party (DUP) and Ireland, but that it had yet to be signed off by any of those involved, the Guardian reported on Thursday. British factions have squabbled about how to preserve the 1998 Good Friday Agreement, which depends on a borderless passage between the Republic of Ireland and Northern Ireland, as Britain seeks new trade independence that would typically require a border.
The border issue has been threatening to derail the divorce talks.
The UK, which is due to leave the European Union in March 2019, wants to open talks on a new free trade deal as soon as possible.
The Government I lead will never be neutral when it comes to expressing our support for the Union.
Foreign Secretary Boris Johnson has insisted any Brexit deal must stick to the spirit of the Leave campaign.