The Prime Minister has been given until midnight on Sunday to put forward a solution to the Irish border deadlock and other initial divorce issues.
If this is not met, Brexit negotiations will not move on to trade talks before the end of the year.
“So far no white smoke,” European Commission spokesman Margaritis Schinas told reporters in Brussels on Thursday.
“We stand ready to receive Prime Minister May at any moment in time when they are ready.”
The DUP torpedoed Mrs May’s efforts to reach an agreement during her trip to Brussels on Monday, with Arlene Foster pulling her party’s support over concerns at the wording of a draft text she feared would agree regulatory alignment between Northern Ireland and the Republic after Brexit.
Mrs May spoke with both the DUP leader and the Republic of Ireland’s Prime Minister Leo Varadkar on Wednesday in her attempt to salvage a deal with the EU.
Her efforts appeared to be further complicated on Thursday when Foreign Secretary and Vote Leave figurehead Boris Johnson warned any Brexit deal must respect promises made during the EU referendum campaign.
“It is very, very important that whatever happens now, whatever we agree, has got to be consistent with taking back control of our laws, of our borders and our cash,” he said.
Without an agreement from EU leaders at next week’s European Council summit, Brexit negotiations will not move on to trade talks before the New Year.
Sky News’ Europe Correspondent Mark Stone said: “This is effectively the third deadline issued by the EU.
“The first loose deadline was the October summit, when both sides hoped the ‘sufficient progress’ test on phase one of the negotiations would be met.
“When that failed to happen, the deadline was shifted to the December summit.
“In order to meet the December summit, the UK was given a 10-day deadline on 24 November. That ran out last Monday. The new deadline is this Sunday.
“In reality, this deadline could be moved too. But there are political limits to what the EU can do because each member state will want to see the draft text first before deciding anything.
“The later the Prime Minister comes, the greater the risk other nations will not sign up to a hastily rushed statement.”
But, despite the difficulties facing the Prime Minister ahead of Brussels’ looming deadline, Transport Secretary Chris Grayling insisted he was “very optimistic” about the hopes of a Brexit agreement.
Speaking to Sky News, the Cabinet minister played down suggestions Mrs May should walk away from negotiations if talks did not progress.
He said: “Nobody should expect us to sign up to a bad deal for Britain, but I’m very optimistic about this.
“I think we want an agreement. All my conversations with other European ministers suggests they want an agreement.
“I’m sure we’ll get there.”
However, Mr Grayling voiced his disagreement with a House of Lords report published on Thursday, which stated that a no-deal Brexit would “deeply damage” the UK.
“Britain’s going to succeed come what may,” the Leave-supporting MP said.
He also appeared to contradict Chancellor Philip Hammond’s suggestion the UK would still hand over a Brexit divorce fee of up to £50bn even if there was no final withdrawal agreement.
Mr Grayling said: “We and the EU have said nothing is agreed until everything is agreed. Until that point there isn’t an agreement to fulfil.”
The Prime Minister has come under pressure from her backbenchers not to leave the UK too closely aligned to the EU after Brexit.
It follows suggestions closer ties with Brussels in some economic areas have been proposed as a means of ensuring there is no hard border on the island of Ireland.
Amid the ongoing discussions between Dublin, Belfast, Brussels and London, Conservative backbencher Bernard Jenkin claimed the Irish Government was “being played like a harp” by the European Commission.
He told BBC’s Radio 4 Today programme: “Dublin is being used as a proxy by the EU.
“The EU is extremely worried about setting the precedent that you can leave the EU and you can have an invisible frontier.”
Mr Jenkin also warned the UK “shouldn’t be allowing ourselves to be bullied into promising more and more money or giving up the goal of regulatory autonomy” after Brexit.
On the other side of the Tory divide on Brexit, prominent Remain supporter Anna Soubry has called for the Government to give MPs the power to seek an extension to negotiations if a satisfactory trade agreement isn’t reached by March 2019.