Theresa May has been given 10 days to offer further concessions on issues including the Brexit divorce bill and the complex matter of the Northern Irish border if she wants European Union leaders to agree to trade talks.

The Prime Minister hopes a crunch summit in Brussels next month will give the green light to move on to the next stage of the Brexit process, covering future trading arrangements and a possible implementation period to avoid a cliff-edge for businesses.

Talks on trade will not be allowed to begin until European Union 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.

After talks with the Prime Minister, European Council president Donald Tusk said it was “possible” sufficient progress could be made at the December summit but remained a “huge challenge”.

“We need to see progress from UK within 10 days on all issues, including on Ireland,” he said.

As she left a gathering of European leaders in Brussels, Mrs May said: “There are still issues across the various matters we are negotiating on to be resolved but there has been a very positive atmosphere in the talks and a genuine feeling that we want to move forward together.”

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.

The Prime Minister 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.

A Number 10 spokesman told reporters on Friday the UK continued to look for “an innovative way forward” on the issue.

Brexit divorce bill(PA Graphics)

Asked whether Northern Ireland could remain in the customs union following Brexit, the spokesman said: “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.”

As well as talks with Mr Tusk and Mrs Merkel in Brussels, the Prime Minister also had meetings with Danish premier Lars Lokke Rasmussen, Belgium’s Charles Michel, and Lithuania’s Saulius Skvernelis in the margins of the Eastern Partnership summit.

One of the issues under discussion was thought to be the amount the UK is prepared to offer Brussels following reports that Cabinet ministers agreed to double the sum originally put on the table by Mrs May to around £40 billion.

However it is thought she 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.