Meanwhile, the EU's chief negotiator Michel Barnier warned that the UK's withdrawal agreement could still fail over the issue of the Irish border, leading to an "extremely serious" no-deal Brexit.
"We are working to ensure that we have that future relationship in place by the end of December 2020", May said as she arrived at European Union headquarters in Brussels on Thursday for meetings on migration, security and other issues.
Britain's Brexit minister, Dominic Raab, said a week ago that the so-called backstop created to prevent a hard border between the British province of Northern Ireland and the Republic of Ireland, must be finite, short and time-limited.
Both leaders spoke by telephone earlier this week, however there were no indications they are closer to resolving the outstanding Irish border issue.
France did not rule out a possible lengthening of Britain's post-Brexit transition period if it helps advance negotiations with London, but any extension would come with conditions, an official in President Emmanuel Macron's office said on Thursday.
An further year could help the United Kingdom and the EU negotiate a future trade deal, and could ease the burden for May on agreeing to a so-called backstop on the Irish border issue that would keep Northern Ireland in the customs union and the single market for goods.
Lidington rejected claims that an extension would cost Britain more than $13.1 billion USA in payments to the European Union in order to stay in the bloc's single market and customs union for another year.
Less than six months before Britain quits the European Union in its biggest shift in policy for more than 40 years, the two sides are at odds over how to deal with their only land border, between the British province of Northern Ireland and Ireland.
May said a proposed 21-month transition period for the United Kingdom after Brexit could be extended by "a matter of months".
The meeting in Brussels was supposed to be the occasion when the leaders of the European Union 27 member states gave the green light for a special summit in November to finalise the terms of Britain's withdrawal.
The Tanaiste said concerns about the potential to destabilise the island's "very precious peace" were real, and reflected the fears of people living on the border.
"But I believe everybody around the table wants to get a deal".
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"I am still cautiously optimistic [about the deal]".
Ahead of her visit to Brussels, Mrs May was able to secure the backing of her Cabinet, at least for now, amid reports that some Brexiteer ministers had been prepared to quit if she gave too much ground to Brussels.
From October 17 through October 18, leaders of the 27 EU states conducted a European Council meeting focused on Brexit.
"We need time to get ready for that".
He also said there would be "huge difficulties" within the European Union 27 member states to consider making the backstop UK-wide, adding that the European Union is willing to make exceptions to treat Northern Ireland as a special case because of its hard history and unique geography.
"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.
If May loses, there would be a leadership contest with the victor becoming prime minister, without the need for new elections.
Such a border would violate the Good Friday Agreement (GFA) or Belfast Agreement that Northern Ireland's present devolved system of government is based on.
"I am open to the idea of an extension or a longer transition period, but that's not an alternative to a legally binding Irish backstop, so it's certainly not an alternative; it's something that might be part of the mix", he said.
Both sides think an extension would allow for more time to find a solution to the Irish border.
Malmstrom said the Europeans "see this merely as preparations being made by the U.S.to negotiate with them and others".