There has been "some progress" towards a Brexit agreement on the Irish border but it remains "50/50" whether a deal will be reached, Sky News understands.
At around midday, the pound hit a day's high of 1.3523 against the dollar after news emerged that Westminster will concede on European Union trade rules for Northern Ireland.
The concession has yet to be accepted by the Irish government.
Theresa May and Jean-Claude Juncker were forced to admit it was not possible to strike a Brexit deal on Monday after a frantic day of negotiations came to a halt because Northern Ireland's DUP had not yet been signed up.
"If the United Kingdom offer is unacceptable for Ireland, it will also be unacceptable for the European Union", he said after meeting Irish prime minister Leo Varadkar.
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London Mayor Sadiq Khan, who called for London to be offered access to the single market if Northern Ireland could have it, tweeted that the "clear solution to this government chaos" was for the whole of the United Kingdom to remain in the single market and customs union, this sidestepping the issue of where the border would be.
Mr Rees-Mogg, a prominent Brexiteer who has been talked about as a future Conservative leader, said the prime minister could not agree to different regulatory systems in Northern Ireland and the rest of the EU.
"We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the U.K.", DUP leader Arlene Foster said.
The Prime Minister is reliant on any support from the party as the DUP is propping up her government with its 10 MPs.
May and Juncker made no comment to reporters when they met at the European Union executive's Berlaymont headquarters for a lunch that diplomats and officials hope can seal a breakthrough that would open the way to negotiations on future trade relations.
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In London, May's spokesman said: "Progress is being made".
May said that "on a couple of issues some differences do remain which require further negotiation and consultation".
"I hope we are in a place this evening where Irish people north and south will get reassurance from the wording that is very close to being finalised now". But the issues of the rights of expatriate citizens and the UK-EU border on the island of Ireland defied a deal until the last minute.
"And the problem between the Republic and Northern Ireland is a outcome of the British decision not only to leave the EU but also the single market and the customs union".
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