The Prime Minister brought up the possibility of an extension during meetings with European Union leaders in Brussels on Wednesday in an effort to get passed a deadlock in negotiations.
That more positive atmosphere was echoed by other European Union officials, saying May appeared to show greater understanding for some of the EU's concerns, including Ireland's need for an insurance "backstop" to avoid a hard border with Northern Ireland - the key stumbling block so far in the talks.
Today, Mrs May will have further bilateral meetings, having had talks on Wednesday with Jean-Claude Juncker, the European Commission President, Donald Tusk, the European Council President, Leo Varadkar, the Irish Taoiseach, and Emmanuel Macron, the French President.
Have made good progress on Brexit talks.
To get over the optics, the language from the European Union included the nicely fudged phrase that they "stand ready" to call a special summit in November to sign off a deal.
"We have shown we can do hard deals together constructively".
Mr Tajani suggested that Mrs May had shown willingness to look into the possibility of extending the proposed 21-month transition period following Brexit to three years.
"I'm afraid she is losing the confidence now of colleagues of all shades of opinion, people who have been supportive of her throughout this process".
Michel Barnier said ‘we need more time’ to reach a deal
And she told them: "The last stage will need courage, trust and leadership on both sides".
Speaking after hearing Mrs May's address to EU leaders, European Parliament president Antonio Tajani said: "There was a message of goodwill and readiness to reach an agreement, but I didn't perceive anything substantially new in terms of content as I listened to Mrs May".
The prime minister and her team are willing to consider the longer transition as a way to overcome the the Irish border issue, according to people familiar with the matter.
A spokesperson for the prime minister refused to categorically rule out an extended transition period earlier on Wednesday, telling reporters in Westminster: "We're not calling for an extension to the implementation deal".
But the idea is controversial among May's eurosceptic Conservative MPs, who fear it is simply a way of delaying Britain's exit.
"This brings with it a whole array of questions, such as: How, the day after Brexit, do we manage the estimated 100,000 British citizens who, in some cases, have been living in Germany for years?"
Asked about the plan, a British official said there were no plans to propose a change to the transition.
Arriving for the talks, he said "we need time, we need much more time" for talks, vowing to work "calmly and patiently" for a deal in the coming weeks.
The Prime Minister travels to Brussels for what had been billed as "the moment of truth" in the negotiations amid growing concerns the two sides will be unable to bridge the gap over the key issue of agreeing a "backstop" deal to ensure there will not be a return to a hard border in Ireland when the United Kingdom quits the EU.
May said this proposal was unacceptable as it would create an array of new customs and regulatory checks between Northern Ireland and Great Britain, consequently undermining the constitutional integrity of the UK.
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