This came as Jean-Claude Juncker, the President of the European Commission, was forced to deny a report that he had described May as "despondent" and "tormented" following a meeting last week.
A leaked account of a private dinner between Juncker and May, published in a German newspaper, claimed that Juncker told colleagues the Prime Minister had requested the meeting as a "call for help" and was under so much pressure that "it takes all her strength not to lose her temper".
The leak was quickly blamed on Juncker's German head of cabinet Martin Selmayr, who was also accused of leaking details of the pair's Downing Street dinner in April, but Selmayr denied the charge.
The leak claimed Juncker had told colleagues May seemed "anxious, despondent and discouraged" and had the appearance of "someone who gets no sleep at night".
Juncker claimed to be shocked by the claims, saying "nothing is true in all of this", adding: "She was in good shape, she was not tired, she was fighting, as is her duty."
Asked if May had begged him for help, he said: "That is not the style of British Prime Ministers".
May today updated Parliament on the Brexit talks following the two-day European Council meeting in Brussels last week, and insisted she had "a degree of confidence" that trade talks would open in December.
However, she made it clear that unless a trade deal was agreed by next summer Britain would have to leave the EU on World Trade Organisation (WTO) terms and scrap the proposed two-year transition period.
May said: "The point of the implementation period is to put in place the practical changes necessary to move to the future partnership.
"In order to have that you need to know what the future partnership is going to be."
Until now, the Government had left open the possibility that trade talks could continue into a transition period.
Barnier suggested he still expects that to be the case, telling France's Les Echoes newspaper that the second phase of talks "will be very different and will last several years".
However in a sign progress is being made, he said his team was "already starting work on a draft of the treaty for the exit of the UK from the EU".
He conceded the future UK-EU trading relationship was the "most important thing".
He warned that a no deal Brexit would have severe consequences.
"We do not wish it at all, we do not work on it, but we do not exclude any option," Barnier said.
"Such a scenario would cause us problems, and even greater ones in the United Kingdom."
He said a no-deal Brexit would mean immediate problems and said the EU could not recognise the qualifications of British pilots or authorise them to land or take off from European airports.
