Update: British Prime Minister Theresa May has said she was "crystal clear" to EU leaders at the Brussels summit about the need for assurances on the backstop in response to MPs' concerns.
Mrs May welcomed commitments by the EU27 to try to get a new trade deal completed quickly enough that the backstop would not be needed, and said that as formal conclusions from the summit they had "legal status".
But she said that it was clear that "further clarification and discussion" is possible.
Mrs May said she had a "robust" discussion with Jean-Claude Juncker after she was asked about footage showing her speaking to the European Commission president.
NEW - Two expert lipreaders tell 5 News that Theresa May accuses Jean-Claude Juncker of describing her as nebulous.
This is how the conversation went, according to the lipreaders: pic.twitter.com/IuP99fJiXG— Channel 5 News (@5_News) December 14, 2018
The Prime Minister said: "I had a robust discussion with Jean-Claude Juncker - I think that's the sort of discussion you're able to have when you have developed a working relationship and you work well together.
"And what came out of that was his clarity that actually he'd been talking - when he used that particular phrase - he'd been talking about a general level of debate."
Mrs May said: "The EU is clear, as I am, that if we are going to leave with a deal, this is it.
"But my discussions with colleagues today have shown that further clarification and discussion following the council's conclusions is in fact possible.
"There is work still to do and we will be holding talks in coming days about how to obtain the further assurances that the UK Parliament needs in order to be able to approve the deal."
Following reports that European Commission president Mr Juncker had described her presentation to EU leaders on Thursday as "nebulous", Mrs May said: "I was crystal clear about the assurances that were needed on the backstop, having heard the views of MPs in the House of Commons."
Mrs May said it was in the interests of the EU and UK to "get this over the line", warning: "A disorderly Brexit would be good for no-one."
She said that the EU27 conclusions made clear that the EU is determined to work speedily on a future relationship - or alternative arrangements - to ensure by the end of 2020 that there will be no hard border in Ireland, so that the backstop need not be triggered.
The conclusions confirm that any backstop would be temporary and that, if it was ever needed, the EU would negotiate "expeditiously" an agreement to put a replacement in place.
She said: "As formal conclusions, these commitments have legal status and therefore should be welcomed."
But she said she had discussed with EU leaders, including Mr Juncker and Council President Donald Tusk, the fact that MPs would require "further assurances".
She said: "It is in the overwhelming interest of all our people in the EU and the UK to get this done and as quickly as possible."
Theresa May is under mounting political pressure after EU leaders rejected her attempts to win fresh concessions to the agreed Brexit deal and openly criticised her negotiating strategy.
The British Prime Minister went to Brussels seeking a way to get the Withdrawal Agreement through a heavily divided Commons, insisting she could do it but had to be able to convince MPs the UK would not find itself tied to the EU indefinitely through the backstop.
She urged EU27 leaders join her to "work together intensively to get this deal over the line in the best interests of all our people".
But after listening to her appeal, European Council president Donald Tusk said EU leaders had reaffirmed their determination that there could be no renegotiation.
He said they had expressed a "firm determination" to work "speedily" to ensure that there was an agreement on the future relationship in place by the time the transition period ends in December 2020, so the backstop was not needed.
European Commission president Jean-Claude Juncker went further and criticised Mrs May's lack of clarity over what she was seeking from the future relationship.
"Our UK friends need to say what they want, instead of asking us to say what we want," he said.
"So we would like, within a few weeks, our UK friends to set out their expectations for us because this debate is sometimes nebulous and imprecise and I would like clarifications."
He added: "We don't want the UK to think there can be any form of renegotiation, that is crystal clear. We can add clarifications but no real changes.
"There will be no legally binding obligations imposed on the withdrawal treaty."
Arriving for the second day of the summit, Mrs May went straight into a one-to-one meeting with French president Emmanuel Macron.
The EU hardball approach appears to leave Mrs May with limited room for manoeuvre during the countdown to the UK's departure on March 29.
The Prime Minister, who on Wednesday survived a bruising vote of no confidence by Tory MPs, said a package of assurances around the backstop could "change the dynamic" at Westminster.
At the same time, she made clear that a failure by EU leaders to offer concessions risked the collapse of the whole agreement, with the UK leaving in March in a disorderly, no-deal Brexit.
She ended with a highly personal appeal to EU leaders to put their trust in her to deliver on her promises and to give her the political room for manoeuvre she needs.
Last night, Taoiseach Leo Varadkar confirmed in the clearest terms that the backstop is “not on the negotiating table” and said “the deal we have is the only deal on the table”.
Asked about the possibility of any concessions, he said: "When it comes to the assurances the prime minister is seeking, the EU is very keen to offer her explanations, assurances, clarifications, anything that may assist MPs to understand the agreement and hopefully support it.
“But the backstop is not on the table. I don’t think we could agree to anything that would change the treaty, change the content.”
Czech Europe Minister Ales Chmelar said the meeting had not been bad but that "nobody knew beforehand what would be the solution".
He told BBC Radio 4's Today programme: "I know there was in the room a strong willingness to offer something, but at the same time it would put, most probably, into question the backstop for Ireland.
"It was almost an impossible situation and technically speaking there was not an easy way out."
Britain's Shadow Brexit minister Sir Keir Starmer said Mrs May had failed to achieve "meaningful changes" and called for a meaningful vote on the deal before Christmas.
But Cabinet Office Minister David Lidington rejected this and defended Mrs May's handling of the talks, telling Today: "Anybody who has heard Theresa May in debate, anybody who has heard her around the Cabinet table, knows there is a very clear plan."
He described the talks as "a welcome first step that was the removal of uncertainty" over the EU's intentions, because it had shown it wanted a "speedy UK trade deal" that would remove the need for the backstop in the first place.
- Digital Desk and PA