He also promised: “The UK will never be an ordinary third country for us. The UK will always be a very close neighbour and partner in political, economic and security terms.”
His remarks, in his annual “state of the union” speech to the European Parliament in Strasbourg, were being seen in Whitehall as a further sign that EU negotiators are ready to engage on the details of Mrs May’s proposals.
They followed reports from Brussels and Berlin that EU leaders want to alter chief Brexit negotiator Michel Barnier’s instructions to allow more room to conclude a deal.
But they also raised concerns among Eurosceptic Tories that the Prime Minister will be forced into further compromises.
Mr Juncker also used his speech to repeat the EU’s “red lines” in the Brussels talks.
He said: “We respect the British decision to leave our Union though we regret it. But we also ask the British Government to understand that someone who leaves cannot be in the same privileged position as a member state.”
Mr Juncker said the European Commission wanted to find “a creative solution” to solving the dispute about the future of the border between Northern Ireland and the Irish Republic.
He added: “It is not the European Union, it is Brexit that risks making the border more visible.
“In the past months, whenever we needed unity in the Union, Britain was at our side, driven by the same values and principles as all other Europeans.
“This is why I welcome Prime Minister May’s proposal to develop an ambitious new partnership for the future, after Brexit. We agree with the statement made in Chequers that the starting point for such a partnership should be a free trade area between the United Kingdom and the European Union.”
Mr Juncker added that EU negotiators were ready “to work day and night to reach a deal”.
Mrs May responded in the Commons by saying: “I believe a deal that is right for the UK will be a deal that is right for the European Union.
“When we leave the European Union, we will be an independent sovereign state but I want to say to our closest allies in Europe: You will also never be an ordinary third party for us.”
Today Exit Secretary Dominic Raab will send another signal that Britain will be prepared to leave without a deal next year if talks fail to make progress.
His department will release around 30 notices setting out measures the Government, businesses and households should take to be ready on issues such as mobile charges, environmental protections and vehicle standards.