British and European Union officials resumed talks to clinch a Brexit deal on October 15 just a few hours after late-night negotiations wound up, but it was far from clear they would reach an agreement before a leaders' summit on October 16.
He also said Prime Minister Boris Johnson would meet the undertakings given to a Scottish court earlier this month, when the government's top legal officer in Scotland gave assurances the government would fully comply with a law which requires Johnson to request a delay to Brexit if no deal is reached by October 19.
By agreeing to a form of customs boundary in the Irish Sea, Britain could allow its province of Northern Ireland to remain under European Union rules, prevent a return to a hard land border with European Union member Ireland and salvage a negotiated withdrawal.
But Prime Minister Boris Johnson may struggle to convince hardline Conservative eurosceptic MPS and his allies from Northern Ireland's loyalist Democratic Unionist Party (DUP) to accept this concession - less than three weeks before Britain is due to leave the EU.
The negotiating teams were continuing to try and reach a deal before a crucial summit on Thursday and Friday.
But current European Union president Finland said more time was needed and that negotiations could continue even after the European Union summit.
But the sides are trying to achieve in a few days what they had failed to in the more than three years since Britons first voted to leave the European Union after almost 50 years.
German EU minister Michael Roth echoed that sentiment, saying he not "quite sure if a deal is close".
Hopes for a Brexit breakthrough have been kept alive after a suggestion from the Irish deputy premier that a deal could be struck as early as this week.
In a statement issued alongside the Queen's Speech, Johnson said the British public were "tired of stasis, gridlock and waiting for change".
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Negotiators went behind closed doors for intensive talks in Brussels after Johnson outlined a new set of divorce terms to Irish Prime Minister Leo Varadkar on Thursday. We've spent three years trying to get an agreement between the two sides and have made progress at different times. However, if a significant political change takes place in the United Kingdom then that could potentially justify a discussion on an extension if we were asked for it, ' France's Amelie de Montchalin said.
The party needs "a deal that respects Northern Ireland's constitutional position as per the Belfast Agreement within the United Kingdom and indeed respects the economic integrity of the United Kingdom single market", Foster said in an interview with Irish broadcaster RTE.
But a small Northern Irish party that supports Britain's Conservative minority government said further work was required to get a deal through. Johnson will be told to seek an extension to European Union membership, but he has sworn not to delay Brexit further and a legal battle will probably follow.
The last and seemingly least likely outcome would see parliament vote on an actual agreement reached in Brussels against all odds this week.
Finland EU affairs minister Tytti Tuppurainen said the EU must prepare for a no-deal and another extension of the tortuous divorce process.
But both sides say gaps remain over plans for keeping goods and people flowing freely across the Irish border, the thorniest issue in the talks.
The alternative is to have checks between Britain and Northern Ireland.
BBC political editor, Laura Kuenssberg, said she understood the two leaders had agreed on the need to avoid any further delays if possible.
The UK proposed a plan that would see Northern Ireland remain in the UK's customs territory but operating the EU's rules and procedures on tariffs.