But European leaders insisted they had made their final offer during months or acrimonious negotiations that resulted in an unpopular compromise for which May ended up paying with her job, AFP reported.
The British prime minister's voice broke on the steps of her Downing Street office when she told Britons on Friday that she was quitting on June 7.
May is bowing out with her legacy in tatters and the country in agony over what to do about voters' decision in 2016 to abandon the European integration project after nearly 50 years.
The markets view the risk of Britain crashing out of the bloc when the twice-delayed departure date arrives on October 31 as uncomfortably high.
The pound has been steadily losing value since May 6 and British business lobbies are raising the alarm.
Their main concern is that current frontrunners to head May's Conservative Party say they will get Brexit done at any cost.
"We will leave the EU on October 31, deal or no deal," said former foreign minister Boris Johnson in a speech delivered Friday in Switzerland.
"The way to get a good deal is to prepare for no deal. To get things done you need to be prepared to walk away."
Johnson's main challenges will come from former Brexit secretary Dominic Raab – viewed as an even more committed eurosceptic – and Foreign Secretary Jeremy Hunt.
Britain's top diplomat had bitterly opposed Brexit in 2016 but has since reversed himself and made headlines in September by comparing the European Union to the evils of the former Soviet Union.
'Enthusiastic about Brexit'
The contest is being held against the backdrop of European Parliament elections that the new Brexit Party of the anti-EU populist Nigel Farage is expected to win with about a third of the vote.
Polls show the Conservatives getting punished for their bickering over Brexit and finishing as low as fifth – their worst result in a national election.
The contenders are also mindful of a party revolt over May's fateful decision to court the pro-EU opposition with the promise of a second Brexit referendum.
The concession was designed to help ram her withdrawal agreement through parliament on the fourth attempt.
But it won her no converts and sparked a party coup attempt that forced May to walk away before she was pushed out.
This prompted more EU-friendly hopefuls such as Work and Pensions Secretary Amber Rudd to concede Saturday that they stood no chance and would not vie for the job.
"I am conscious the Conservative Party wants someone who they believe is very enthusiastic about Brexit," Rudd told The Daily Telegraph.
Johnson is a popular figure viewed by many Conservatives as the party's best answer to Farage.
But a long political career that also saw him serve as London's cosmopolitan mayor has made him enemies in parliament who will try to block his rise to the top.
Parliamentary party members will begin whittling down the field of contenders to a final two on June 10.
The finalists will go up for a vote in a ballot held across Britain by around 100,000 party members with three months' standing in July.
The field grew on Saturday when Health Secretary Matt Hancock entered the race with a promise to take a more moderate approach.
Leaving the European Union without an agreement is "not an active policy choice that is available to the next prime minister," Hancock told Sky News.
Hancock is view as one of the dark horses who might make it through a crowded field that is expected to include more than a dozen names.
International Development Secretary Rory Stewart is also positioning himself as a more consensus seeking alternative to Johnson.
"It now seems that (Johnson) is coming out for a no-deal Brexit," Stewart told BBC radio.
"I think it would be a huge mistake. Damaging, unnecessary, and I think also dishonest."
Yet neither Hancock nor Stewart would say if they would push ahead with May's current agreement or try to secure added concessions from Brussels.
Dutch Prime Minister Mark Rutte – seen as one of Britain's better allies in Europe – told reporters Friday that "the withdrawal agreement is not up for renegotiation".