The EU's chief negotiator Michel Barnier says there are "fundamental divergences" in positions when it comes to the matter of securing the rights of EU citizens living in the UK post-Brexit.
The EU insists the European Court of Justice should be the top court dealing with the issues but the UK disagrees.
The first full round of negotiations are wrapping up in Brussels after beginning on Monday.
The UK's minister for leaving the European Union David Davis - who took part in the joint press conference - says they don't expect a so called "punishment deal" and that the UK wouldn't accept one.
The EU is still seeking clarification from the UK on its positions regarding the border between the Republic of Ireland and Northern Ireland after Brexit.
And the amount the UK will be expected to pay the EU in outstanding financial commitments is still unresolved.
Here's the UK Brexit minister David Davis.
IN: "We agree..."
OUT: "...this week."
DUR: 21 secs
CLIP1: http://www.fsnradionews.com/feeds/0720brexit-davis1.mp3
ALT
The EU chief Brexit negotiator Michel Barnier called on the UK to provide clarification on many of the issues already being discussed.
IN: "Our objective..."
OUT: "...British position."
DUR: 27 secs
CLIP: http://www.fsnradionews.com/feeds/0720brexit-barnier1.mp3
ALT
The UK Brexit minister David Davis explained comments from his colleague in the UK cabinet, Dr Liam Fox, about the suggestion that the EU would punish the UK during the negotiations.
IN: "He said if..."
OUT: "...good deal."
DUR: 9 secs
CLIP1: http://www.fsnradionews.com/feeds/0720brexit-davis2.mp3
ALT
The EU chief Brexit negotiator Michel Barnier called on the UK to provide clarification on many of the issues already being discussed.
IN: "Our objective..."
OUT: "...British position."
DUR: 27 secs
CLIP: http://www.fsnradionews.com/feeds/0720brexit-barnier1.mp3
ALT
The UK Brexit minister David Davis explained comments from his colleague in the UK cabinet, Dr Liam Fox, about the suggestion that the EU would punish the UK during the negotiations.
IN: "He said if..."
OUT: "...good deal."
DUR: 9 secs
CLIP1: http://www.fsnradionews.com/feeds/0720brexit-davis2.mp3