Talks on finalising the UK's withdrawal agreement and a political declaration on future trade relations with the EU have been blocked by failure to agree on a backstop to avoid a hard border between Northern Ireland and the Republic in the event that an agreement on future EU-UK relations can not be struck.
Her difficulties were laid bare by the fury sparked back home at her suggestion that she could extend a post-Brexit transition period to address the fraught issue of the Irish border.
The idea of an extension had been floated by the EU's chief negotiator Michel Barnier.
TRT World's Simon McGregor-Wood reports from Brussels. Theresa May has now confirmed that an extension is an option being considered.
May has repeatedly ruled out extending what she calls an implementation period, but at least one of her ministers has indicated he may support such a step.
At present, the two sides say Britain will remain inside the EU's vast single market, and subject to the bloc's regulations, from the day it leaves until December 2020, to give time for new trade relations to be set up. More of them are supporting a call for another referendum on European Union membership, on the grounds that parliament, as well as the ruling party, are hopelessly divided. Some have described the idea as "desperate" and "completely ridiculous".
Tajani said May said she would evaluate an idea to extend the post-Brexit transition period by a year from the current cut-off date at the end of 2020 but that the British prime minister did not say whether she would favour that, or not.
Mrs May has argued the backstop should see the whole of the United Kingdom stay in a customs union with the EU on a temporary time-limited basis, after the transition period comes to an end.
Lithuanian President Dalia Grybauskaite said that 2 ½ years after Britain's Brexit referendum, the country had still not explained clearly how it wants to leave the EU.
But she failed to respond to a call by EU President Donald Tusk for "concrete proposals" of her own to move the talks forward. "If it did help to reassure people that the backstop would never be activated, that would be a positive thing", said Irish Prime Minister Leo Varadkar.
She said: 'We do not know what they want, they do not know themselves what they really want - that's the problem.
The UK government's plans would create a separate system for goods which might mean checks and controls, bringing disruption and a potential return to the sectarian troubles of the past.
Prime Minister Theresa May (left) and Arlene Foster, the leader of the Democratic Unionist Party (DUP) during a visit to Belleek pottery factory on the northern side of the border between Enniskillen in Northern Ireland and Ballyshannon in Donegal.
Many suspect that will not be enough time, which has led the European Union to demand a "backstop" to ensure there are no customs posts or other controls along the now invisible border between Northern Ireland and Ireland.
A soft Brexit would also largely solve the Northern Ireland dilemma, which now threatens to wreck the entire Brexit process, and risks triggering the break up of the United Kingdom.
The EU side is still insisting that the exit treaty include its proposed backstop.
"The last stage will need courage, trust and leadership on both sides", May told her European Union counterparts.
LONDON - The last few weeks of the Brexit process must have felt like the closing stages of a particularly brutal chess game for Theresa May. Mr Varadkar told reporters that a regular summit had been fixed for December and clear progress would be required to show the value of a special one next month.