Britain is due to leave the European Union on March 29, but so far the government has not been able to win Parliament's backing for its divorce deal with the bloc.
The left-of-centre party said it would back a second public vote if the House of Commons rejects its alternative Brexit plan, which calls for Britain to retain close economic ties with the EU.
Voters narrowly approved the June 2016 referendum to pull Britain out of the EU.
Labour MP David Lammy, who backs a "People's Vote" on the terms of the UK's exit, welcomed the move by his party to "back the principle" of a referendum.
Speaking to Reuters, Asselborn noted that a second referendum would likely shift the Brexit timetable back by six months or more, from the current March 29th date.
Labour has not until this point definitively backed a second public vote on Brexit - preferring instead to push for another general election.
However, a party source insisted that Labour was taking anti-Semitism extremely seriously and was constantly strengthening its internal processes.
Labour is not yet making clear what its proposed referendum would be on.
Chuka Umunna, one of the eight MPs who quit the party to join TIG, said the announcement was "better late than never".
At the moment though, and after his talks with her on Sunday, Tusk said that "Prime Minister May still believes that she is able to avoid this scenario" of extending the departure beyond March 29.
May wants to change the deal to reassure British lawmakers that the backstop would only apply temporarily.
Three EU officials said they would be ready to approve a short Brexit delay if Britain needed more time to ensure parliamentary ratification of their divorce agreement.
The EU has been watching with growing concern the political machinations in Britain and the possibility that the country will crash out of the bloc without a deal, risking chaos on both sides of the Channel.
The Bank of England also forecast in November that a no-deal Brexit in March would cause the United Kingdom economy to shrink by about 8% within a year, the worst drop the country has seen since the 1920s.
In a dramatic statement to the House of Commons, Mrs May confirmed that she will put her Withdrawal Agreement - including whatever additional assurances she has secured from Brussels - to a "meaningful vote" by March 12.
"An extension of the Article 50 (departure) deadline could still result in a no-deal outcome", said Sarah Hewin, chief Europe economist at Standard Chartered. "What kind of message would that send to the more than 17 million people who voted to leave the European Union almost three years ago now?" Ms.
"I don't see how businesses can plan, I don't see how public services can plan and I think it's just deeply damaging", Cooper told the BBC.
Ms James, who is considered one of the Government's rising stars, said: "We don't know what the whipping is going to be. what the Prime Minister is going to say in Parliament today".
He said "all the 27 (member states) will show maximum understanding and goodwill" to make possible such a postponement - a decision that would require a unanimous vote from them.
Just over two-thirds of the government's most critical preparation projects for a no-deal scenario - and fewer than 85% overall - were "on track" for completion in time for the scheduled date of Brexit on March 29, it said. Despite urging from within her own party, the option of a delay is something May had dismissed. "It doesn't resolve the issue".
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