"And, we should be clear that as for now, not enough progress has been made".
Leaders reaffirmed their confidence in Barnier, seen as a snub in the face of the UK's efforts over the summer to go behind the European Union negotiator and make deals with member states instead.
Tajani said May told him she would consider it but gave no indication of whether she favoured such an extension.
However, a fraught weekend in Brussels culminating in UK Brexit Secretary Dominic Raab walking away from the negotiating table means a deal won't be signed off until November - if there is a deal at all.
"What has now emerged is the idea that an option to extend the implementation period could be a further solution to this issue of backstop in Northern Ireland".
British Prime Minister Theresa May is weighing up a plan to stay tied to European Union rules for longer in a radical move created to break the deadlock in Brexit talks, according to people familiar with the matter.
They added: "We've always said that the transition should be as short as possible but as long as necessary".
The comments come as London is nearing a March 2019 deadline to leave the EU.
May mentioned the possibility of prolonging the transition immediately after saying the leaders had made "real progress" - something she has said repeatedly in recent days.
In the event of a "no-deal Brexit", any transition period plans are liable to disintegrate.
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As it's hard to see the brooch in this picture, we're hoping the Palace releases another photo of Eugenie where it is in frame. A group photograph taken in the White Drawing Room at Windsor shows both the bride and groom's families.
The idea of a one-year extension to the transition period had been proposed by EU Brexit negotiator Michel Barnier.
The UK's biggest trading partner is the European Union, providing a market for about 44% of all British exports in goods and services, and supplying more than half of its imports. Any deal agreed in Brussels needs to win approval in Parliament, where May doesn't have a majority and faces opposition on all sides.
One MP called for the Prime Minister to "stand aside" and five former Tory cabinet ministers issued an open letter urging her to scrap her strategy.
However, an extension does not in and of itself solve May's problem, which is that she has yet to fully decide, let alone agree, what sort of relationship she wants to secure during that additional transition period.
He conceded "we are not so far" from a deal but cautioned that "now we must accelerate the work".
"The Prime Minister and her Government are too weak and too divided to protect people's jobs, our economy or ensure there is no hard border in Northern Ireland", he said.
So uncertain was that prospect that a special European Union summit on Brexit that had been penciled in for next month to finalize a deal was taken off the table.
The prime minister failed to reassure MPs from the Democratic Unionist Party that there'd be no new regulatory checks between Northern Ireland and the rest of the United Kingdom after Brexit, and received a kicking from numerous pro-EU MPs, including up to 10 MPs who urged May to back another referendum.
British Euro-skeptics oppose Britain remaining in a customs union with the European Union, which is May's ultimate goal, because they say it would limit Britain's ability to strike trade deals around the world and force it to abide by rules it will have no power to shape. However, she dodged questions from pro-Brexit MPs like Boris Johnson and Iain Duncan Smith, who asked her whether the backstop would have a fixed end date. "If they knew the negotiations were being pushed right up against the next election, why would they be conciliatory in these negotiations?"
Tusk broke off the meeting, after nobody responded to the British Prime Minister.