An EU official signaled that Donald Tusk, the chairman of EU leaders, could be willing to offer even longer: up to a year for Britain's feuding politicians to agree and ratify a plan. France, however, indicated it was not yet ready to accept an extension unless the British presented a clear plan which would justify such a delay.
"We're not there today," a source close to French President Emmanuel Macron told Reuters.
Britain is now due to leave the EU in a week, but May has been forced to seek more time after Britain's parliament failed to approve a withdrawal agreement, Presstv Reported.
Her Conservative Party is deeply divided, as is the main opposition Labour Party, leading to an extraordinary series of inconclusive votes in parliament that have stretched Britain's centuries-old unwritten constitution to its limits.
Scenarios that run the gamut from abandoning the EU abruptly with no exit deal to cancelling Brexit altogether have all gone down to defeat.
Obscure parliamentary procedures have been resurrected from the rulebooks providing daily drama from the House of Commons, but the future of Britain's biggest change in generations has become no clearer.
After finally recognizing that her minority Conservative government could not push through a Brexit deal on its own, May started talks this week with Labour leader Jeremy Corbyn in the hope of coming up with a cross-party solution.
But that means accepting the need for more time, including the prospect that Britain might have to hold European Parliament elections on May 23, which May has long said she hoped to avoid at all cost.
"The United Kingdom proposes that this period should end on 30 June 2019," May said in the letter.
"The government will want to agree a timetable for ratification that allows the United Kingdom to withdraw from the European Union before 23 May 2019 and therefore cancel the European Parliament elections, but will continue to make responsible preparations to hold the elections should this not prove possible."
Tusk, who convenes a summit of EU leaders next week, is likely to offer Britain a flexible extension of up to a year, with the possibility of leaving sooner, a senior EU official said.
"The only reasonable way out would be a long but flexible extension. I would call it a 'flextension'," the official said.
As in May's proposal, the extension could be terminated early if Britain ratifies the withdrawal agreement. "It seems to be a good scenario for both sides, as it gives the UK all the necessary flexibility, while avoiding the need to meet every few weeks to further discuss Brexit extensions," the official said.
Any extension must be agreed by all 27 of the other EU countries. France in particular has signaled that it would not automatically give Britain whatever May sought.
"If we are not able to understand the reason why the UK is asking for an extension, we cannot give a positive answer," Finance Minister Bruno Le Maire told reporters in Bucharest.
The French diplomatic source called the extension idea premature and "clumsy".
However, other European politicians have signaled they would be happy to give Britain time to rethink.
Armin Laschet, the premier of Germany's biggest regional state, North Rhine-Westphalia, tweeted: "If Britain asks for an extension to avoid a chaotic exit from the EU with incalculable risks for hundreds of thousands of jobs, we should agree." He added: "The longer the better. That means the Brits take part in the European elections too."
British parties divided
May offered to quit last week to get her deal passed but it was defeated for a third time last Friday, the day Britain was originally due to leave the EU. The EU had given her an extension until April 12 and said it could be extended to May 22, but only if parliament agreed the withdrawal deal.
Her latest gamble on talks with the Labour Party has infuriated the pro-Brexit wing of her Conservatives and divided her cabinet.
Labour too is divided. It is officially committed to leaving the EU but with closer ties than May has sought, including a customs union, which May has so far ruled out.
Many Labour members and lawmakers also want to put any agreement to a second public vote - potentially opening a path for Brexit to be rejected altogether. Party leader Corbyn has been difficult to pin down on whether this would still be necessary if May agrees to a customs union.
Corbyn's deputy Tom Watson, who supports a second referendum, said it would be difficult for Labour to back any agreement without it.
"We're genuinely going in with an open mind, but if it comes out of that process without the idea of a confirmatory ballot, I think we would have a bit of difficulty with our parliamentary party," Watson told BBC radio.