- Video report by ITV News Correspondent Angus Walker
Theresa May has been given 10 days to offer further concessions on issues including the Brexit divorce bill and the border between Northern Ireland and the Republic of Ireland, if she wants EU leaders to agree to trade talks.
Yet speaking in Brussels after fresh talks with European Union leaders at a summit in the Belgian city, Mrs May described their discussions as "positive", but conceded that there are "still issues across the various matters that we are negotiating on to be resolved".
European Council President Donald Tusk said agreement to post-Brexit trade talks in December was "still possible" but remained a "huge challenge" and progress from the UK within the next 10 days was needed.
If sufficient progress is not made by the UK, then trade talks will not be allowed to begin until EU leaders are satisfied that "sufficient progress" has been made on the first round of issues being discussed including the divorce bill the UK will pay to Brussels and the Northern Irish border.
On the border issue, Mrs May is coming under intense pressure from Dublin for fresh assurances there will be no "hard border" between Northern Ireland and the Republic, with Taoiseach Leo Varadkar warning that deadlock in Brexit negotiations cannot be broken until the issue is resolved.
Speaking after her discussions, Mrs May insisted "we have the same desire - we want to ensure that movement of people and trade across that border can carry on as now".
Mrs May's comments came after Downing Street backed away from suggestions that Northern Ireland's continued membership of the EU customs union could be up for negotiation in Brexit talks.
On Friday, a Number 10 spokesperson said the UK continued to look for "an innovative way forward" on the issue.
Asked whether Northern Ireland could remain in the customs union following Brexit, they added: "That is a matter for negotiations."
But a Downing Street source later insisted the Government's position that the whole of the UK will leave both the customs union and single market after Brexit has not changed.
Irish foreign minister Simon Coveney said on Friday it was difficult to see how border checks could be avoided if the UK's departure from the customs union and single market resulted in "regulatory divergence" between the North and the Republic.
Mr Varadkar has previously suggested a "bespoke" arrangement, similar to that operated on the Isle of Man, under which Northern Ireland, or the whole of the UK, would continue to observe the rules of the single market and customs union without necessarily remaining a member of them.
The EU's chief Brexit negotiator Michel Barnier discussed the situation with Mr Coveney in Brussels ahead of the European Council meeting on December 14-15 which will decide whether talks can progress as Mrs May hopes.
Mr Barnier said there was "strong solidarity" with Ireland, adding that "Irish issues are EU issues".
The Prime Minister said: "In relation to the border between Northern Ireland and the Republic of Ireland, we and the Irish government continue to talk about the solution to that.
"But we have the same desire - we want to see that movement of people and trade across that border can carry on as now and that we don't create any new barriers to trade or the movement of people across that border.
"That's the outcome that we are both agreed on and that is what we believe is in the best interests of Northern Ireland."
In other meetings on the margins of the Eastern Partnership summit, the amount the UK is prepared to offer Brussels is thought to have been discussed.
Discussions on the so-called divorce bill follow reports that Cabinet ministers agreed to double the sum originally put on the table by Mrs May to around £40 billion.
But it is thought the Prime Minister does not want to name a precise figure until she has a clear idea of what kind of trade deal is available with the remaining EU member states in the phase two negotiations.
The Prime Minister said: "We have been talking about how we can progress the issue in relation to the financial settlement. I have set out the position. I did so in the Florence speech.
"I said that we would honour our commitments. I said that no member state of the European Union need worry that they would receive less or have to pay more in the current budget plan."