European Council President Donald Tusk touches his eyebrow after delivering a statement during a joint news conference following a Tripartite Social Summit roundtable at the European Council headquarters in Brussels, Tuesday, Oct. 16, 2018.
Meeting Juncker was the first of several one-on-one meetings on May's schedule before she addressed her 27 European Union counterparts on bringing the negotiations over Britain's departure to completion.
Merkel said: "The chance of achieving a good and sustainable withdrawal agreement in good time is still there, and it is really in the interest of our relations with Britain, in the interest of our economy. and of course in the interest of people in our countries".
The EU is also insisting on its own "backstop", which would see Northern Ireland alone stay aligned with the bloc's customs union and single market - something Britain will not accept.
Former Brexit minister Steve Baker said this week that even that was a "hideous implementation period".
Because of its timing, the summit in Brussels was billed long in advance as the "moment of truth" in the two-year Brexit process.
Leaders have not yet decided on whether to call for an extra summit in November in order to close the deal, or to discuss further preparation plans in case an agreement is not reached.
Lithuanian president Dalia Grybauskaite claimed the Prime Minister didn't have a "strong mandate". The EU says a time limit would defeat the point of the backstop.More news: Astros possibly caught cheating in ALCS Game 1
Her official spokesman later told reporters: "The Cabinet strongly supported her on the importance of maintaining the integrity of the Union and agreed that we must ensure we can not be kept in a backstop arrangement indefinitely".
"The original transition was an unnecessary trap created by our weak civil servants who can not be trusted as they don't want us to leave. That is the problem".
That means leaders will only decide whether enough progress is being made to hold an emergency summit in November or whether a deal will have to wait until the European Council in December. She told reporters that "most of the issues" in the withdrawal agreement have been resolved - apart from the Irish border.
Last December, EU and UK negotiators agreed to a 21 month transition period after the UK formally leaves the bloc in March next year during which the UK would remain part of the single market but have no say over new rules governing it, and on the need to agree a backstop on the Irish border.
An further year could help the United Kingdom and the EU negotiate a future trade deal, and could ease the burden for May on agreeing to a so-called backstop on the Irish border issue that would keep Northern Ireland in the customs union and the single market for goods.
"There have been a lot of reports lately, mostly on social media, that my department has been planning to lower food and farming standards when negotiating Free Trade Agreements post-Brexit", Fox will tell farming industry representatives in a speech.
Some Conservatives are already upset at the idea of the transition itself, which is now due to end in December 2020.
The EU official also said May had told leaders that Britain was prepared to consider an extension.
But both London and Brussels have pledged to avoid any physical infrastructure, or "hard border", anxious that it could upset the delicate peace process that ended decades of violence between Protestant supporters of British rule over the province, and Irish Catholics nationalists, who believe in a united Ireland.