An accord to ensure a smooth British exit from the European Union is still possible but the risk of a no-deal is also "very real", European Commission President Jean-Claude Juncker has said.
In an address in the European Parliament, Mr Juncker said British Prime Minister Boris Johnson had told him on Monday that London still wanted a transition deal, but that he would leave with or without an accord on 31 October.
Pro-Brexit MEPs cheered and applauded in the Strasbourg chamber.
"There is very little time left...The risk of a no-deal is very real," said Mr Juncker, who met Mr Johnson in Luxembourg on Monday.
In a worst case scenario, a no-deal Brexit could mean severe disruption to trade, supplies of medicines, fresh foods and a possible rise in public disorder, according to the British government's contingency plans.
Such a sharp break in economic ties, ending four decades of EU membership, "might be the United Kingdom's choice, but never the choice of the EU," Mr Juncker said, highlighting how the bloc wants to avoid blame if Britain crashes out.
Mr Juncker said London must present realistic proposals to replace the backstop in the Withdrawal Agreement, which former Prime Minister Theresa May agreed with EU leaders but which was rejected by the British parliament.
"I am not emotionally attached to the Irish backstop," Mr Juncker said. "I have asked the prime minister to make, in writing, alternatives," he said, calling it a safety net.
The backstop is designed to prevent the reintroduction of a hard border with customs checks on the island of Ireland after Brexit.
He told MEPs that his talks with Mr Johnson on Monday had been "friendly, constructive and in part positive".
"The commission is prepared to work day in day out, morning until night - with a few breaks - to try to find the technical and political solutions we need but I am not sure that we will get there," he said.
"There is very little time remaining but what I do know is that we have to keep trying."
Mr Juncker's pessimistic tone was echoed by Finland's minister for European affairs, Tytti Tuppurainen, who also spoke in the parliament, saying a no-deal Brexit "is a quite likely outcome." Finland holds the EU's rotating presidency.
However, many MEPs warned against a no-deal, both to avoid an economic shock and because they do not want to see Britain abandon its commitments to EU social and environmental standards and become a low-tax, low-regulation rival.
"We will not accept a Singapore on the North Sea," said former Belgian prime minister Guy Verhofstadt, a liberal EU lawmaker and a member of the parliament's Brexit committee.
In at times bad-tempered debate underscoring general weariness on the tortured issue of Britain's pending departure, senior EU lawmakers took other jabs at the noisy contingent of British eurosceptic deputies in the chamber.
Manfred Weber, leader of the centre-right EPP group, called Brexit "stupid", while he and Mr Verhofstadt took aim at British plans for greater sovereignty at a time when the parliament in Westminster has been suspended by Mr Johnson.
"Brexiteers claimed Westminster would take back control, but now they shut it down," Mr Weber, a German MEP, said.
Brexit Party leader Nigel Farage said it was clear from the "emollient" comments by Mr Juncker that they were "very close" to a deal at the EU summit next month in Brussels.
However, he said such an agreement would be bad for the UK as it would leave it reliant on the "good faith" of the EU when it came to negotiating the future relationship, when it was clear from recent events that there was no such good faith.
"I am of course referring to the pipsqueak prime minister of Luxembourg who set out to ritually humiliate a British prime minister in the most astonishing way, only to greeted as a hero by President [Emmanuel] Macron at the Elysee Palace," Mr Farage told the parliament.
"The only way forward now is to deliver on the referendum is for a clean break Brexit.
"Once we have done that we will have a grown-up conversation about trade and about the way forward."
The European parliament is due to adopt a resolution later today calling for Britain to be granted another extension to allow more time for London to agree the terms of its withdrawal.
EU leaders will discuss whether to grant an extension at a two-day summit in Brussels from 17 October. Britain's departure has already been delayed twice since March and Mr Johnson says this will not happen again.
The EU's chief negotiator, Michel Barnier, said that a no-deal Brexit would also not resolve many of the issues around the rights of EU citizens, the Irish border and British obligations under the bloc's long-term budget.
"If the United Kingdom leaves without a deal, all these questions will not disappear. They are still there," Mr Barnier said. "Some three years after the Brexit referendum we should not be pretending to negotiate."
Additional reporting PA