The Prime Minister told a crunch European Council summit in Brussels that she was "ready to consider" the extension floated by EU chief negotiator Michel Barnier to give time to resolve the intractable problem of the Irish border.
Tusk says "what I feel today is that we are closer to the final solutions and the deal".
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.
If agreed, the change would mean the United Kingdom remaining within the single market and customs union and subject to EU rules and regulations for nearly three years after the official date of Brexit in March 2019 and more than five years after the referendum vote to Leave.
On Thursday, British Prime Minister Theresa May and other European Union leaders voiced renewed confidence that they could secure a Brexit deal, yet the two sides remain at odds over how to deal with their only land border, between Northern Ireland and the Republic of Ireland.
Britain has secured a 21-month transition period following the formal date of Brexit in March 2019, to give authorities and companies time to prepare for new arrangements.
May's official spokesperson said, "We've shown we can do hard deals together constructively".
Varadkar also warned that a return of customs posts on Ireland's border with Northern Ireland could lead to violence returning to the British province.
Arriving for the second day of the European Council summit, Mrs May made clear she would accept an extension only as a means to ensure there was no hard border in Ireland if it proved impossible to implement the future partnership by the end of 2020. Under the current proposal, the United Kingdom would effectively be a non-member: it would remain part of the single market and customs union, but would no longer be represented in EU institutions.
The Taoiseach met with British Prime Minister Theresa May for half an hour last night before the main Brexit discussion with all 28 countries, including the UK.
Jeremy Corbyn, leader of the main opposition Labor Party, in a statement on Thursday said: "We are leaving the European Union, but we will not support a deal cobbled together by a divided and chaotic Conservative government if it's going to make life tougher for millions of people".
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He added: "Leo Varadkar has taken a sledgehammer to the work which was done but even more worrying is his total disregard to the impact of his irresponsible and reckless rhetoric on the peace of Northern Ireland".
I understand Theresa May is prepared to agree to another year of Brexit transition (membership of European Union stripped of voting rights) to 31 Dec 2021 so long as the European Union abandons an "unacceptable" Northern Ireland-only backstop of customs and single-market membership.
"There is a certain difference in the assessment of where we are now", admitted an European Union official of the conclusions reached by the leaders at the end of their two-hour dinner.
Tory MP Nadine Dorries repeated her call for former Brexit secretary David Davis to replace Mrs May as leader.
And that is tricky politically for May because her government relies on the party's support.
"As long as we don't have a solution we won't be able to explain exactly how it can succeed", she said Thursday.
Mrs O'Neill said: "In order to preserve her toxic alliance with the DUP, she is trampling over the rights of citizens by acquiescing to that party's refusal to share power (at Stormont) on the basis of equality".
"That's very much a decision for the British people and the British government", he said.
The possibility of extending the transition period, set to end by December 2020, was also mooted at the Brussels summit. "They do not know themselves what they really want".
The idea of extending the transition would be to give more time to negotiate a deal on future relations and find a formula to defuse the Irish border question.