REPORTS from Brussels suggested that Mr Juncker warned a summit of leaders from the remaining EU 27 on Saturday that the Brexit negotiations could collapse because of the differences of the two sides on key issues.
They followed Wednesday's meeting at No 10 – which was also attended by the EU' s chief negotiator Michel Barnier – which was said to have gone "very badly".
Afterwards, Mr Juncker was said to have telephoned German chancellor Angela Merkel to warn her that Mrs May's approach to the negotiations was from a "different galaxy" to the other member states.
His comments prompted Mrs Merkel to issue her own warning of "illusions" held within the UK that achieving a deal would be straightforward.
Mrs May sought to play down the reports, saying they simply showed that the forthcoming negotiations would be "tough" at times.
However they were seized upon by opposition parties who said the government was heading for a "hard Brexit" which would leave people worse off.
Labour shadow Brexit secretary Sir Keir Starmer said: "Whatever the purpose of these leaks, this is a deeply worrying account and further evidence that Theresa May's rigid and complacent approach to Brexit negotiations risks leading Britain over a cliff edge.
"Theresa May talks about strengthening her hand, but in reality she has misjudged her hand at every turn, weakening Britain's position. By refusing to acknowledge the complexity and magnitude of the task ahead the Prime Minister increases the risk that there will be no deal, which is the worst of all possible outcomes.
"In pursuing a rigid and complacent approach, the prime minister now finds herself marginalised and isolated across the continent. Since day one, she has been driven not by the national interest, but by the interests of the Tory party."
Liberal Democrat Leader Tim Farron said: "It's clear this government has no clue and is taking the country towards a disastrous hard Brexit Theresa May chose a divisive hard Brexit, with Labour's help, and now has no idea what to do next."
For the SNP, Scotland's minister for UK negotiations with the EU Michael Russell said: "This looming catastrophe shows precisely why Theresa May mustn't be allowed to block the decision of the Scottish Parliament to give people in Scotland a choice over their future when the terms of Brexit are clear.
"Leaving the EU with no deal - and no agreement on access to the single market - would be an unprecedented act of self-harm which would devastate the UK and Scottish economy.
"It is absolutely essential that we have a strong opposition at Westminster who can hold this kamikaze Tory leader to account and put the Brexit talks under real scrutiny."
No 10 said that it did not recognise the account of the meeting which appeared in the German Frankfurter Allgemeine Zeitung newspaper.
A Government spokesman said: "As the Prime Minister and Jean-Claude Juncker made clear, this was a constructive meeting ahead of the negotiations formally getting under way."
