European Council President Donald Tusk has warned Britain there will be no withdrawal agreement and no transition deal unless the problem of the border between Ireland and the UK is solved.
Speaking during a debate in the European Parliament in Strasbourg, Mr Tusk said: "The UK’s decision on Brexit has caused the problem, and the UK will have to solve it."
"Without a solution there will be no withdrawal agreement and no transition."
He added that he hoped "positive momentum" in Brexit talks can help to deal with the outstanding problems.
European People’s Party leader Manfred Weber said Ireland would be supported during the Brexit talks.
"We are all Irish in this regard and we stand clearly behind our Irish friends," he said.
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DUP MEP Diane Dodds said the deal reached on Brexit was an important staging post, but far from perfect.
She said her party wanted to ensure that there will be no barriers within the UK’s internal market and called for technological solutions to the border problem, rather than the idea of a special status for Northern Ireland.
Commenting on the Brexit deal, Fine Gael MEP Mairead McGuinness said the idea that there would be no withdrawal agreement and no transition deal was a very stark prospect for the UK.
She urged those with doubts about the problem to visit the border area, and listen to the people involved.
"Tragically, in the referendum you forget about Northern Ireland, but Europe has not forgotten", she said.
UKIP MEP Nigel Farage said the UK did not vote for a transition deal, it voted to leave the EU.
He said he was concerned the UK is heading for a "really bad deal from a British perspective", which would include "open door migration, the main reason we voted to leave the in the first place.
"I’m pretty frustrated, and I think 17 and a half million people are too."
Mr Farage said, however, that he was pleased the attempt to hold a second referendum had run into the ground and the UK would leave the European Union.
"Once we are a sovereign country again we will be able to unpick this mess."
Elsewhere, the UK government has suffered its first defeat in the House of Lords over the European Union (Withdrawal) Bill after peers voted in favour of a customs union amendment by 348 votes to 225.