Mrs May faced a Tory backlash after signalling she is ready to delay the UK's final departure from the EU's hold until 2021 in a last ditch bid to end the deadlock over the Irish border issue.
And the suggestion - first made by European Union negotiators - that Britain could stay firmly in the EU's sphere for longer than the 21 months expected has fired up those who already say Britain has already given up too much in the negotiations.
"We need much more time and we will continue to work in the next weeks, calmly and patiently", he added.
Previously, both sides had agreed that Britain crashing out of the Union on March 29 next year with neither a divorce agreement nor a road-map to future ties would be an economic and diplomatic disaster.
The lack of progress means a special European Union summit on Brexit that had been penciled in for next month has been scrapped, though European Union leaders said they would assess the situation later. It is understood that the meeting was "cordial and positive".
The group called on Mrs May to "reset" the negotiations and ditch her Chequers Brexit blueprint in favour of a Canada-style free trade agreement.
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Barnier said on Friday that failure to resolve that issue could still derail any deal.
While staunch Brexiteer Nadine Dorries called for former Brexit secretary David Davis to step in as "interim leader" over the possibility of an extended transition period, Tory backbencher Johnny Mercer described the current administration as a "shit show" in a fiery interview on Thursday.
Because her chief whip Julian Smith presumably has advised her that if she were to adopt the additional year of what the Brexiters see as pure vassalage - without simultaneously scrapping the backstop and Chequers - there would be (in the words of one) "the mother of all explosions".
One prominent Brexit supporter, Conservative lawmaker Jacob Rees-Mogg, told Reuters that any extension to the transition was "a bad and expensive idea".
Britain's Prime Minister Theresa May arrives for the European Union leaders summit in Brussels, Belgium October 18, 2018.
He added: "In March this year we were told "oh there's going to be an implementation period and you are stuck shadowing the CFP until the end of 2020".
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Mrs May relies on her DUP allies' 10 MPs to prop up her minority government in key votes and there have been suggestions the Northern Irish party could vote against the Budget. Efforts to allay these objections, centred on fears of splitting the North from the rest of the United Kingdom, have run into trouble with the EU side who want to protect the single market and customs union.
A senior Democratic Unionist has branded Ireland's premier "vile" after accusing him of using terrorist victims to scaremonger over Brexit.
"My view is that we should really only have one if there is a objective to having one. We have been able to negotiate changes and on foot of them we had a second referendum in the past but I do recall Irish people and me personally taking offence at foreign leaders coming over to intervene on our referendums so I would never repeat that mistake".
Speaking to BBC Radio 4's Today programme, Mr Hunt said: "The reason why this week has been hard is because Theresa May has not buckled".
"Today, we do not know what they want".
At a three-hour cabinet meeting on Tuesday morning, which included ministers with reservations about her strategy, May said a deal was possible if they all stood together. Earlier in the day, Theresa May had told reporters of the great progress that had been made.
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But for now the absence of a breakthrough will be less important to the prime minister than the absence of a breakdown. The EU is expected to send a letter to these partners asking that existing trade arrangements continue intact until the end of the transition; in other words, that they continue to treat the United Kingdom as party to these agreements even though it will have formally left the EU.
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