The future U.K. -EU border in Ireland has been the main sticking point in the talks.
EU Brexit chief Michel Barnier has warned that a deal could fall at the final hurdle if a solution isn't found to the Irish border issue.
"And we should be clear that, as for now, not enough progress has been made", he said.
But later on Thursday, she attempted to play down the comments and said Britain will not be pushing for an extension to the Brexit transition period.
While leading Brexiteer Jacob Rees-Mogg described the move as a "poor attempt at kicking the can down the road", pro-EU MP Nick Boles said it would be "madness". "Such a regime should help to protect our citizens, companies and institutions from all kinds of cyber security threats", he said.
"The answer is yes", Barnier said when asked on France Inter radio if the Irish border issue could cause the negotiations to collapse. "I believe we need a deal".
"A further idea that has emerged, and it is an idea at this stage, is to create an option to extend the implementation period for a matter of months, and it would only be a matter months", May told reporters on the second day of the summit.
That more positive atmosphere was echoed by other European Union officials, saying May appeared to show greater understanding for some of the EU's concerns, including Ireland's need for an insurance "backstop" to avoid a hard border with Northern Ireland - the key stumbling block so far in the talks.
Tory Brexiteers have been told to "concentrate their fire" on Theresa May instead of her officials by former Whitehall mandarins as tensions over talks with the European Union mount. While it could help May sell a deal at home, it won't remove the need to accept the most controversial part of the EU's proposal - that in a last-resort situation, Northern Ireland could be split off from the rest of the United Kingdom.
"We are all working, we're intensifying the work on these issues that remain", May told a news conference after a two-day summit in Brussels had ended.
It now depends on whether May can sell such an extension to the contingent of strong Brexiteers among the Conservative benches as well as to the 10 MPs from the Democratic Unionist Party who shore up her minority government.
European leaders had hoped to hold a special summit in November to seal the divorce, but at a Brussels dinner without May late Wednesday they refused to sign off on the plan.
They concluded that the progress in the EU-UK Brexit talks was not sufficient in spite of multiple negotiations on the matter.
Yet if a deal is to be struck, the focus will be finding a formulation that makes a customs border between Northern Ireland and the British mainland highly improbable and/or makes a UK-EU future customs zone more certain than Brussels now wants.
At dinner with European Union leaders on Wednesday, Mr Varadkar circulated a newspaper article on the Newry bombing. Efforts to allay these objections, centred on fears of splitting the North from the rest of the United Kingdom, have run into trouble with the EU side who want to protect the single market and customs union. "They do not know themselves what they really want".
As leaders from different political traditions in Britain we have disagreed about much in the past, but what we have in common- our support for democracy, the rule of law, free expression and free trade - are also fundamental European values and they are all worth defending.
On the Irish border issue Tajani said, "We want a flexible border and to preserve the Good Friday Agreement, but we also want to protect our agri-food sector, industry and health".
With divorce talks stuck, the bloc has suggested extending that period, to give more time to strike a trade deal that ensures a frictionless border between Northern Ireland and Ireland.