Chief EU negotiator Michel Barnier said "we need much time, much more time, and we continue to work in the next weeks".
"They are trying to leave this so late that they can credibly say there is no alternative but a "no-deal" Brexit, and most people agree that would be chaos", Boles told the BBC. "But I have to say the period since I made that speech we have had a very different approach from the European Union a more constructive approach that has led to the situation we're in now where all but one or two issues have been resolved".
British Prime Minister Theresa May had earlier addressed her fellow leaders, indicating that she could accept extending the post-Brexit transition phase to take the heat out the deadlocked issue of the Irish border.
An accord sketched by officials from both sides last weekend stalled because May's fractious government rejected a "backstop" insurance clause the European Union wants in case future talks fail to forge a trade pact that avoids customs posts on the Irish border.
"People who have been supportive of her throughout this process, they are close to despair at the state of this negotiation", said Conservative lawmaker Nick Boles, a critic of May who during the referendum backed remaining in the European Union and now wants to stay in the single market on an interim basis.
The transition period, which would come into force when Britain officially leaves the EU in March 2019, is meant to be a temporary arrangement to continue the current UK-EU relationship while a future trade deal is negotiated.
Leaders have not yet decided on whether to call for an extra summit in November in order to close the deal, or to discuss further preparation plans in case an agreement is not reached.
Tom Newton Dunn, political editor of the Sun newspaper, commented that May saying a transition extension would only be for a few months feels like the "I'm only half pregnant" defense.
UK Prime Minister Theresa May.
Extending the transition period could mean that if a future partnership is not ready, a backstop, which so far has been unpalatable to the British side, would not have to be triggered.
Syed Kamall, the leader of the Conservative's group, is "quite confident that there will be a deal by the end of the year", a spokesperson said on Friday.
Juncker and May, however, remain hopeful that the Brexit deadlock could be broken and an agreement reached.
She is hemmed in by pro-Brexit members of her Conservative Party, who oppose any more compromises with the bloc, and by her parliamentary allies in Northern Ireland's Democratic Unionist Party, who insist a solution to the border issue can't include customs checks between Northern Ireland and the rest of the United Kingdom.
Former Ukip leader Nigel Farage said that it would delay full withdrawal nearly until the general election scheduled for May 2022 and "may mean we never leave at all". Barnier emphasised the "unanimity on the part of the EU" that there should be no closed border between Northern Ireland and the Republic of Ireland.
Simon Collins, the Executive Officer of the Shetland Fishermen's Association, warned the Government it should not let fishermen down again - insisting the key issue between the fishing industry and the UK Government remains to be "trust" in Brexit talks.
At present the two sides say Britain will remain subject to the bloc's rules from Brexit day on March 29 until December 2020, to give time for new trade relations to be set up.
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.
Its co-chair Richard Tice said: "The original transition was an unnecessary trap created by our weak civil servants who can not be trusted as they don't want us to leave".
Varadkar also warned that a return of customs posts on Ireland's border with Northern Ireland could lead to violence returning to the British province.
Both sides think an extension would allow for more time to find a solution to the Irish border.