Brussels has warned Brexit talks will stall unless Britain sets out details of its willingness to pay a divorce bill and demanded clarification on a range of issues after four days of negotiation broke up.
The EU's chief Brexit negotiator insisted all accounts "must be settled" when the UK quits the bloc and said he was not ready to compromise in negotiations.
Michel Barnier said "fundamental" disagreements remain over citizens' rights, arguing that reforms must be underpinned by European courts.
Brexit Secretary David Davis said talks had been "robust but constructive" and urged Brussels to show "flexibility".
Labour said the lack of progress in the second round of negotiations was "deeply concerning" and did not bode well for the future.
At a press conference in Brussels, Mr Barnier said: "There does remain one fundamental divergence on the way in which such rights would be guaranteed and on several other points, for example, the rights of future family members or the exports of certain social benefits."
Further details about the UK's willingness to pay a fee to Brussels will be required before talks can move on to a trade deal, Mr Barnier indicated.
But the UK is understood to think the EU team is being unclear on what it believes the legal obligations are over the divorce bill as well, with frustration on both sides.
Mr Barnier said: "A clarification of the UK position is indispensable for us to negotiate and for us to make sufficient progress on this financial dossier, which is inseparable from the other withdrawal dossiers.
"What we want - and we are working on this - is an orderly withdrawal for the United Kingdom, that's decided. An orderly withdrawal means accounts must be settled.
"We know that agreement will not be achieved through incremental steps. As soon as the UK is ready to clarify the nature of its commitments, we will be prepared to discuss this with the British negotiators."
Mr Barnier said the first round of talks had been about organisation, this week had been about presentation, and the "third round must be about clarification".
He said: "We require this clarification on the financial settlement, on citizens' rights, on Ireland - with the two key points of the common travel area and the Good Friday Agreement - and the other separation issues where this week's experience has quite simply shown we make better progress where our respective positions are clear."
Mr Davis struck a more optimistic tone, saying: "Overall I'm encouraged by the progress we have made on understanding each other's positions."
He said the talks had demonstrated the UK had made a "fair and serious offer" on citizens' rights and there were "many concrete areas where we agree, as well as areas where there will be further discussion", which will be a priority in the next round.
On the financial settlement, Mr Davis said: "We both recognise the importance of sorting out the obligations we have to one another, both legally and in a spirit of mutual co-operation."
In a sign of the difficulties in reaching agreement he added: "We have had robust but constructive talks this week. Clearly there's a lot left to talk about and further work before we can resolve this.
"Ultimately getting to a solution will require flexibility from both sides."
Mr Barnier said the EU was not ready to compromise in the negotiations until the UK accepts its financial obligations.
He said: "I know one has to compromise in negotiations but we are not there yet.
"When I say, and I think I was very clear and transparent about that, that there are things that are inseparable from others. That's the financial settlement, let's be very clear.
"We want clarity on that because we need to be able work more until we come to areas of compromise."
Underlining his position on the "fundamental importance" of citizens' rights being protected by EU law and the European Court of Justice, Mr Barnier said: "This is not a political point we are making, it's a legal one.
"Simply, if there is to be continuity of EU law, that has to be framed by case law of the court. Only the court can interpret EU law.
"It's not a choice, it's an obligation."
Shadow Brexit secretary Sir Keir Starmer said: "The lack of progress in these negotiations is deeply concerning and does not bode well for the future. On issues such as the rights of EU citizens in the UK and UK citizens in Europe, Labour has been calling for action since July last year.
"Twelve months later, after having been promised a swift resolution of the issue, we learn today that there remain 'fundamental divergences' on this key issue. This means more waiting and more anxiety for millions of families."