Mrs May told leaders of the remaining European Union member states in Brussels that she was ready to consider an extension by "a matter of months" of the transition period, which is now due to stretch until December 2020.
Straight-talking Dalia Grybauskaitė confirmed there was no "moment of truth" in Brussels as the negotiations hit another stalemate. On Thursday the Democratic Unionist Party (DUP), that is in coalition with May's Conservative Party, threatened to remove their support for the government over Brexit if an open-ended backstop in Northern Ireland - an insurance plan to prevent the return of a hard border on the island of Ireland - is to be agreed.
Cabinet Office minister David Lidington, the prime minister's de facto deputy, admitted the cost of extending the transition period would have to be "teased out" during negotiations.
"The Taoiseach wanted to ensure that fellow European leaders fully realise what is at stake and the importance of preserving the peace process throughout the Brexit process", he said.
Citing May's reported preference for an extension of the Withdrawal Agreement transition period, Tajani told reporters, "Reference has been made to three, not two, years and we are in favour of this".
At a dinner on Wednesday night between European Union leaders, UK Prime Minister Theresa May is thought to have suggested a third transitional year after the UK leaves the European Union next March to try to resolve the deadlock over the Irish border issue.
Insufficient progress has been made to convene the European Council to discuss Britain's exit from the EU, Donald Tusk, president of the council, said in a news conference on Thursday.
However, Brexiteers have reacted with anger at suggestions the UK's stay within the EU's structures could be lengthened beyond what had previously been agreed as a 21-month period.
While the EU would like Northern Ireland to remain inside the customs union until a permanent solution is agreed, the UK's proposal would see the United Kingdom as a whole stay in a customs plan with the EU as a temporary measure.
But, despite the absence of an agreement, there were more positive noises from key players compared to the acrimonious Salzburg summit in September.
"The negotiations are very much about a solution and we have got to the point where they have become very technical but certainly it's something that is in the back of my mind as a potential outcome if we get this wrong", he said.
Chancellor Angela Merkel's spokesman said that members also discussed cybersecurity, migration, and the continued development of the Eurozone. Boles added: "They are close to despair at the state of this negotiation", suggesting the PM is complicit in running the clock down for a "No deal" Brexit scenario.
The prime minister remained bullish of a Brexit deal coming eventually.
Simon Hart, founder of the 80-strong Brexit Delivery Group of Tory MPs who want to give Mrs May space to get a good deal, said critics undermining the Prime Minister were "weakening our negotiating position".
Speaking after Tusk and Juncker, May said Thursday that there would be more hard moments ahead as they reach the final stages of the talks, but added that she was confident in her ability to secure a good deal that is agreeable to all parties.
Meanwhile, May was criticised by Sinn Fein president Michelle O'Neill for refusing to meet her party or the SDLP or Alliance to discuss issues related to Brexit.
"Theresa May must now act in the national interest, not her party interest, and break the deadlock by delivering a deal that protects jobs and living standards".