Europeans ask, in effect, a commitment to the maintenance of Northern Ireland within the customs union if London and the 27 did not find other solution to avoid the recurrence of a border visible.
The EU is insisting on a "backstop" clause in any withdrawal treaty to avoid erecting border posts between the British province of Northern Ireland and EU member Ireland if London and Brussels can not agree a trade pact for the future.
EU Council chief Donald Tusk has delivered a tough riposte to Conservative rhetoric over Brexit while holding out the offer of a deal that would deliver "close and special" relations.
UKIP MEP Nigel Farage said the prime minister should "bite his hand off".
The UK government has previously said it was "disappointed" by the decision by Court of Session judges to refer the case to the ECJ, and was giving their ruling "careful consideration".
The prime minister says she is opposed to such a deal precisely because it would require a customs border in the Irish Sea, and would lead to the break-up of the United Kingdom as we know it because Northern Ireland would have to be treated differently to the rest of the UK.
They want to achieve this by agreeing a new trade deal with the continuation of open borders.
The chairman of European Union leaders, Donald Tusk, on Thursday said a "Canada +++" was on offer, meaning an advanced free trade agreement coupled with close security ties and tight cooperation on global affairs, among others.
He also warned that it could take longer than the length of the post-Brexit transition period, due to run to 31 December 2020, to agree on a final EU-UK trade deal.
But speaking with BBC Radio 4's Today programme this morning, the PM said she is ready to present her new version of Chequers to Brussels.
Mr Juncker was quoted as saying 'the rapprochement potential between both sides has increased in recent days'.
But after meeting with May, the Democratic Unionist Party's Arlene Foster stated that "no border in the Irish Sea will ever be acceptable to unionists throughout the UK". If it wants to strike a deal, it is "going to have to screw someone".
The summit on October 18 to 19 has been billed as the "moment of truth" when it will become clear whether it is possible for the two sides to do a deal.
"We are in the art of the possible here, and from what I see in government, I think that we will get a deal, be it in October or November at the two consecutive summits", he said.