The cautious message comes after Mrs May saw her Brexit deal torpedoed by her DUP parliamentary partners – who revolted over the wording of a draft agreement on the Ireland border.
Now EU sources have told the Prime Minister that if she does not secure a deal by Friday then it will stand little chance of being confirmed at the December 14-15 Council meeting.
If the UK is unable to ease the DUP’s worries and win back their support in time for the EU to agree that sufficient progress has been made to move onto trade talks by next week the wait will go on for several months.
More vital time would be lost with the UK likely to have to wait until March for the next time the 27 other countries currently in the EU will unite for a summit.
German finance minister Peter Altmaier said in The Times: “This is what we have been worried about for months.
“We are in the middle of a very, very complicated process. It is very difficult.”
The DUP dramatically appeared to block the Brexit deal after the draft of an agreement between the Irish and UK Governments was leaked to the press.
Arlene Foster and her party were reportedly furious over the wording of the agreement which they believe implied the possibility of legislative divergence between the Northern Ireland and the rest of the UK.
The issue of the future of the border in Northern Ireland has emerged in recent weeks as a new roadblock to Brexit finally moving talks on to a trade deal.
The Irish government, the DUP and Theresa May have all called for a borderless future between the Republic and Northern Ireland.
But the DUP are insistent on there being no border or legislative divergence between Northern Ireland and the UK.
DUP leader Arlene Foster said: “We note the speculation emanating from the European Union exit talks regarding the Republic of Ireland and United Kingdom border.
“We have been very clear. Northern Ireland must leave the EU on the same terms as the rest of the United Kingdom.
“We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom.
“The economic and constitutional integrity of the United Kingdom will not be compromised in any way.
“Her Majesty’s Government understands the DUP position. The Prime Minister has told the House of Commons that there will be no border in the Irish Sea.
“The Prime Minister has been clear that the UK is leaving the European Union as a whole and the territorial and…