All remaining European Union 27 members must agree on any extension.
But a long extension would infuriate the pro-Brexit wing of May's divided party and would require Britain to participate in May 23-26 elections for the European Parliament.
Speaking to the German radio station DRF, Mr Juncker said: 'We will probably have to meet again next week, because Mrs May has not got agreement for anything either in her Cabinet or her Parliament.
EU Council chief Donald Tusk, who is overseeing the summit, said a short delay should be possible, but only if Britain's Parliament approves May's divorce deal with the EU before the scheduled March 29 departure date.
They agreed to extend the Brexit date until May 22, on the eve of the EU Parliament elections, if May can persuade the British Parliament to endorse her Brexit divorce deal.
French president Emmanuel Macron said only a short "technical" extension was on offer and if MPs reject the agreement "it will guide everybody to a no-deal for sure".
He added: "An extension needs to be linked to something new, there needs to be a new event, a new political process". I sometimes have the feeling that we are in the waiting room, a bit like Waiting for Godot.
'I came to office on a promise to deliver on that verdict.
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"On April 12 we have to know where things stand. if we don't have a response by then we will have a no deal Brexit", Luxembourg's Prime Minister Xavier Bettel warned.
"Getting the deal through next week in a meaningful vote means we can have that extension until May 22nd. leave the European Union and do it an orderly manner".
"I think that an extension is better than a no-deal scenario and so for that reason she (British Prime Minister Theresa May) has our support, but we will have to discuss today how long the suspension should be", Kurz told reporters as he arrived at a meeting of EU leaders in the European People's Party - the EU's parliament largest group.
The televised message was described as a "low blow" by former minister Sam Gyimah.
The pound fell sharply as May requested her extension.
"Parliament could vote to revoke Article 50, which is cancelling the Brexit process - I think that's highly unlikely", he said.
"It should not be brought back for a third time of asking".
The European Council meeting will be held on Thursday and Friday, with the Brexit delay expected to be one of the key talking points. On this basis, and in the light of the outcome of the European Council, I intend to put forward a motion as soon as possible under section 13 of the Withdrawal Act 2018 and make the argument for the orderly withdrawal and strong future partnership the United Kingdom economy, its citizens' security and the continent's future, demands.
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The US military's mission is focused on guiding and aiding Afghan forces battling the Taliban, who were ousted from power in 2001. It comes as talks between the US and Taliban militants drag on in a bid to reach a peace agreement.
The vote and a series of political crises since have exposed deep divisions and has fuelled soul-searching about everything from secession and immigration to capitalism and British identity.
According to the Prime Minister, she was still discussing with the European Union any possible amendments to the deal's arrangements for the Irish border.
He continued: "Although Brexit fatigue is increasingly visible and justified, we can not give up seeking until the very last moment a positive solution".
A European Commission document seen by Reuters said the delay should either be several weeks shorter, to avoid a clash with European Parliament elections in May, or last at least until the end of the year, which would oblige Britain to take part in the elections - something that May said was not in Britain's interest.
"Any choice that takes us beyond the European elections would cause serious harm and it would cause serious legal problems", Tajani said.
But Foreign Secretary Jeremy Hunt said she was simply expressing frustration and tapping into the public's exasperated mood.
She added: "I sincerely hope we can do that with a deal".
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The prime minister of New Zealand again visited Christchurch and addressed a massive gathering in Hagley Park to show her empathy. Since then, New Zealanders have responded with an outpouring of support for the country's small Muslim community.