Confronting Mr Tusk over his “Hell” outburst, Mrs May accused the European Council president of undermining efforts to get a deal and triggering “widespread dismay” in the UK.
But British officials last night said signs were growing that the EU was ready to begin serious talks on how to resolve the outstanding issues in the stalled discussions.
One said: “Everyone in the room wanted to get this sorted as soon as possible. They do not want things to drag on any more than we do.”
Tonight, Mrs May will step up her bid for a legally-binding pledge that the UK cannot be trapped in an EU customs union to protect the open Irish border.
She will hold talks with Irish premier Leo Varadkar in Dublin. And Attorney General Geoffrey Cox will fly to the city to meet his Irish counterpart Seamus Woulfe on legal details of the “backstop” border proposal.
Mr Cox is “working on working on possible ways in which legal texts could be drafted, that gave effect to the objective we want,” Cabinet Office minister David Lidington told the BBC.
Following her Brussels meetings, Mrs May said she and EU Commission president Jean-Claude Juncker agreed that “talks will now start to find a way through this, to find a way to get this over the line, and to deliver on the concerns that Parliament has, so that we get a majority in Parliament”.
She added: “What I see and hear from leaders is a desire for us to work together to ensure that we can deliver the UK leaving the European Union with a deal.
“My work is to deliver Brexit, to deliver it on time, and I will be negotiating hard in the coming days to do just that. And I’m clear that I’m going to deliver Brexit, I’m going to deliver it on time, that’s what I’m going to do for the British public.”
Confirming that she gave Mr Tusk a dressing down, the Prime Minister said: “I’ve raised with President Tusk the language that he used, which was not helpful and caused widespread dismay in the United Kingdom. The point I made to him was that we should both be working to ensure that we can deliver a closer relationship between the United Kingdom and the European Union in the future, and that’s what he should be focusing on.”
Hailed Mr Tusk sparked fury during a Brussels press conference on Wednesday by saying leading Brexit campaigners deserved damnation for failing to draw up a viable withdrawal plan.
But even after the rebuke from Mrs May, Mr Tusk was downbeat, tweeting: “Still no breakthrough in sight. Talks will continue.”
Downing Street insiders hailed the decision to open talks between British negotiators and Mr Juncker’s officials as “important progress”, despite EU leaders’ claims that the Withdrawal Agreement could not be renegotiated.
The Prime Minister’s spokesman said: “What we would look to as positive is the fact that there are now going to be talks between the parties to try to find way through which Parliament can support.”
Brexit Secretary Stephen Barclay will be meeting EU chief negotiator Michel Barnier on Monday. In a joint statement after their meeting last night, Mrs May and Mr Juncker said: “The talks were held in a spirit of working together to achieve the UK’s orderly withdrawal from the EU. The discussion was robust but constructive.”
Mrs May and Mr Juncker are expected to hold further talks before the end of the month.
But European Parliament president Antonio Tajani, who met Mrs May yesterday, sounded a warning about a no-deal Brexit.
He said: “We are open to being more ambitious on our future rela-tions, including looking at the Irish situation again if the UK’s red lines change. We are weeks from an economic and human catastrophe. “This is the reality of a no-deal Brexit. The European Parliament has done its part and we need to continue to work together to avoid it.”
The European Parliament’s Brexit co-ordinator Guy Verhofstadt said Mrs May “assured us that there will be a backstop”.
He added: “There is no question to remove the backstop because that is absolutely necessary for securing and safeguarding the Good Friday Agreement, safeguarding the internal market and safeguarding also the peace process.” Mr Verhofstadt said any problems with the backstop should be solved in the Political Declaration.
Support Meanwhile, Mrs May gained fresh support last night from German chancellor Angela Merkel.
Speaking during a visit to Slovakia, Mrs Merkel said: “I think we can find solutions without reopening the Withdrawal Agreement. That is not on the agenda for us.”
The Prime Minister is expected to put the deal to a vote in the Commons towards the end of February.
Bank of England governor Mark Carney has warned of the mounting risk of a recession amid the “negative shock” of a no-deal Brexit.
He said yesterday uncertainty about the outcome of negotiations was “weighing more heavily” on business activity as he slashed growth forecasts to their weakest for 10 years.