But other European Union nations warned this may not resolve the impasse, which has raised fears of the potentially catastrophic scenario of Britain crashing out in March without any agreement.
Former Ukip leader Nigel Farage said that it would delay full withdrawal nearly until the general election scheduled for May 2022 and "may mean we never leave at all".
Pro-EU politicians, meanwhile, said the proposal was another sign of May's weak bargaining hand and an attempt to stall for time.
But the two sides disagree on how to resolve the issue. "Will it not be an issue of principle once we have coughed up billions more in United Kingdom taxpayers' cash?"
The pound fell to intraday lows in late Thursday trade, as summit talks in Brussels failed to resolve a Brexit standoff between London and the European Union over the status of the Irish border, an issue playing an increasingly dominant role in negotiations.
With just 160 days to go to the official date of Brexit, Mrs May urged the EU27 to find a "creative" way out of the current dilemma.
Chief EU negotiator Michel Barnier said "we need much time, much more time, and we continue to work in the next weeks".
"I am confident that we can achieve that good deal".
She goes on to acknowledge "the very last stages of the talks are going to be the hardest of all".
Brussels and London have vowed to prevent the re-emergence of a hard border between the Republic of Ireland and Northern Ireland, which will leave the European Union with Britain, amid fears the issue could reignite decades-old tensions.
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 on the basis of equality".
"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.
Irish Prime Minister Leo Varadkar said a longer transition period was not a substitute for a concrete agreement over the backstop.
When May first requested the transition period, she said it should be limited to a period of about two years.
Antonio Tajani, the president of the European Parliament, said May offered them "nothing substantially new" to discuss.
Leave-supporting cabinet ministers Penny Mordaunt and Michael Gove also signalled discomfort at the prospect of an extension to the transition period.
This week's summit had been billed as "the moment of truth" when agreement was needed to allow time for ratification in the Westminster and European parliaments.
EU Council president Donald Tusk said "not enough progress" had been made to schedule a summit in November, adding to uncertainty over whether a Brexit deal could be reached before time runs out.
But in an article for The Sun newspaper, the prime minister plays down the personal pressures she faces, writing that "the Brexit talks are not about me or my personal fortunes".
Mr Macron, who held separate talks with Mrs May ahead of the three-hour dinner, said it was time to "accelerate" talks.
'If you have a negotiator who has no strong mandate, it's very hard to negotiate'.
Second, the Prime Minister spoke about the Northern Irish backstop, which is meant to guarantee that - in the unlikely event that there is a delay in bringing into effect our future relationship by the end of the implementation period at the end of December 2020 - there is no return to a hard border between Northern Ireland and Ireland.