Ursula von der Leyen spoke with the prime minister over the phone on Thursday evening, when the pair “took stock” of negotiations, which have been continuing throughout the week up against a new deadline.
But there was no sign of a breakthrough as she warned that bringing the divide, particularly on fishing, would be “very challenging”.
A Downing Street spokesperson said the prime minister had underlined during the conversation that talks were “now in a serious situation” and that it “now looked very likely that agreement would not be reached unless the EU position changed substantially”.
Mr Johnson was said to have branded the EU position on fisheries in particular as “simply not reasonable” and said the UK had moved to accommodate the bloc on the level playing field, even though “some fundamental areas remained difficult”.
UK chief negotiator David Frost echoed his boss, stating: “The situation in our talks with the EU is very serious tonight. Progress seems blocked and time is running out.”
On Thursday morning, EU chief negotiator Michel Barnier told senior MEPs that a deal was possible over the next few days, though UK sources have been consistently less upbeat and say there are still key differences in major areas.
The latest top-level conversation took place via telephone, in contrast to an in-person visit by the prime minister to Brussels last week. Negotiations between the two teams of officials are however continuing in person in the Belgian capital.
It comes as the leaders of the European Parliament say they will not meet to ratify a Brexit trade deal this year unless one is signed before Sunday.
They made the comments after a meeting with EU chief negotiator Michel Barnier in Brussels on Thursday morning, in which he is said to have told them a deal was possible by Friday.
Dacian Cioloș, leader of the parliament's liberal Renew group and an ally of Emmanuel Macron, said the deadline was necessary because “uncertainty hanging over citizens and businesses as a result of UK choices” was becoming “intolerable”.
Manfred Weber, chair of the parliament’s largest group, the centre-right European People’s Party, said: “After [Sunday] we cannot reasonably scrutinise the deal before the end of the year. The agreement is too important to rush through Parliament.
“We owe it to the people and businesses in our constituencies who will be heavily affected by Brexit, to scrutinise the deal appropriately. After Sunday we don't believe this would still be possible.”
Iratxe Garcia Perez, leader of the centre-left socialist group, said: “We will not be rushed into a consent vote of the Brexit deal before the end of the year if we don’t have access to the text by Sunday.”
The warning by the Parliament raises the possibility that the UK could be hit by a limited period of trading on no-deal terms from the beginning of January, even if an agreement is signed before the end of the year.
While the European Union and UK both have some powers for “provisional” application of deals without the consent of their parliaments, it is not clear whether these would help prevent a no-deal in this case.
Mr Barnier is understood to have told ambassadors and MEPs in separate private meetings that even the reduced length of the provisional application process means there could well be a short no-deal in January even if it is used.
Ms von der Leyen has also promised the Parliament that she would not use the procedure to bypass MEPs. The joint statement by the parliamentary group leaders recalls this, noting that Ms Von der Leyen said provisional application could take place “only once the European Parliament has given its consent”.
The Independent additionally understands that the European Council’s legal service believes that provisional application would not be fast enough to prevent a no-deal on 1 January if the process was set in motion after this week.
Mr Barnier struck an upbeat tone on the possibility of a free trade agreement on Thursday, saying there had been “good progress” but that “stumbling blocks remain”.
He said talks were in their “final stretch”, adding: “We will only sign a deal protecting EU interests and principles.”
Back in Westminster, Downing Street would not be drawn on the European Parliament’s timetable. The prime minister’s spokesperson said the House of Commons would not be recalled on Christmas Day or the Boxing Day Bank Holiday for its own ratification procedures, but did no rule out weekend sittings to approve the agreement.
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